Analyzer

ABSTRACT

An analyzer includes a magnetic moment application unit configured to apply a magnetic moment to a particle system defined in a virtual space, a magnetic field calculation unit configured to calculate a magnetic physical quantity related to the particle system including particles, to which the magnetic moment is applied by the magnetic moment application unit, and a particle state calculation unit configured to numerically calculate a governing equation, which governs the movement of each particle, using the calculation result in the magnetic field calculation unit. The magnetic field calculation unit numerically calculates an induction magnetic field using induced magnetization induced in each particle due to a time variation in an external magnetic field and a magnetic field obtained by interaction between magnetic moments based on the induced magnetization.

RELATED APPLICATIONS

Priority is claimed to Japanese Patent Application No. 2013-228374,filed Nov. 1, 2013, the entire content of which is incorporated hereinby reference.

BACKGROUND

1. Technical Field

The present invention relates to an analyzer which analyzes a particlesystem.

2. Description of Related Art

In recent years, with the improvement of computational capability of acomputer, a simulation which incorporates magnetic field analysis isfrequently used in the field of design and development of electricappliances, such as a motor. The use of the simulation can improve thespeed of design and development since the simulation enables a certaindegree of evaluation without actually producing a prototype.

For example, Japanese Unexamined Patent Application Publication No.11-146688 describes a motor analyzer including an arithmetic processingunit which executes magnetic field analysis. The arithmetic processingunit executes torque calculation according to magnetostatic fieldanalysis by a finite element method or according to a Maxwell stressmethod in response to external instructions based on user operations.Meshing is performed in the magnetostatic field analysis by the finiteelement method. The meshing is applied to a core region and a housingregion, as well as an external air layer region. Other methods ofmagnetic field analysis include a difference method and a magneticmoment method.

SUMMARY

An analyzer according to an embodiment of the invention includes amagnetic moment application unit configured to apply a magnetic momentto a particle system defined in a virtual space, a magnetic fieldcalculation unit configured to calculate a magnetic physical quantityrelated to the particle system including particles, to which themagnetic moment is applied by the magnetic moment application unit, anda particle state calculation unit configured to numerically calculate agoverning equation, which governs the movement of each particle, usingthe calculation result in the magnetic field calculation unit. Themagnetic field calculation unit numerically calculates an inductionmagnetic field using induced magnetization induced in each particle dueto a time variation in an external magnetic field and a magnetic fieldobtained by interaction between magnetic moments based on the inducedmagnetization.

An analyzer according to another embodiment of the invention includes amagnetic moment application unit configured to apply a magnetic momentto a particle system defined in a virtual space, a magnetic fieldcalculation unit configured to calculate a magnetic physical quantityrelated to the particle system including particles, to which themagnetic moment is applied by the magnetic moment application unit, anda particle state calculation unit configured to numerically calculate agoverning equation, which governs the movement of each particle, usingthe calculation result in the magnetic field calculation unit. Themagnetic field calculation unit numerically calculates an inductionmagnetic field in a particle group using an expression for an inductionmagnetic field derived by gauge transformation using a local gravitycenter vector representing the gravity center position of the particlegroup to be analyzed.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram showing the function and configuration of ananalyzer according to an embodiment.

FIG. 2 is a data structure diagram showing an example of a particle dataholding unit of FIG. 1.

FIG. 3 is a flowchart showing an example of a sequence of processing inthe analyzer of FIG. 1.

FIG. 4 is a diagram showing the calculation results corresponding to anexpansion to n=1 order.

FIG. 5 is a diagram showing the calculation results corresponding to anexpansion to n=2 order.

FIG. 6 is a diagram showing the calculation results corresponding to anexpansion to n=3 order.

FIG. 7 is a diagram showing the calculation results corresponding to anexpansion to n=4 order.

FIG. 8 is a diagram showing the relationship between the orders ofspherical harmonics and calculation values.

FIG. 9 is a diagram showing a hysteresis curve.

FIG. 10 is a diagram showing an analysis model of a conductive spherehaving beads.

FIG. 11 is a graph showing calculation values of a time-varying magneticfield on a conductive sphere surface.

FIG. 12 is a graph showing a phase error and an amplitude error ofcalculation values.

FIGS. 13A to 13C are graphs showing calculation values of a magneticfield on a conductive sphere surface, current density, and anelectromagnetic force.

DETAILED DESCRIPTION

In the related art methods, the meshing is applied to an object to beanalyzed. However, when the object to be analyzed is a fluid and if afixed mesh is used with moving boundaries, the analysis is difficult. Ifthe object to be analyzed is accompanied with large displacement of amagnetic body or a plastic body, the meshing is difficult. Inparticular, when the meshing is performed in magnetic field analysisincluding a rotor, such as a motor, polymerization of meshes (if thepolymerization is single, this is the large displacement) makes theanalysis difficult.

It is desirable to provide an analysis technique which enables preferredanalysis of magnetic phenomenon in a simulation.

It is to be noted that any arbitrary combination of the above-describedstructural components or rearrangement of the structural components andthe expressions of certain embodiments of the invention among a device,a method, a system, a computer program, a recording medium having acomputer program recorded thereon, and the like are also effective asthe embodiments of the invention.

Hereinafter, the same or similar structural components, members, andprocessing shown in the respective drawings are represented by the samereference numerals, and overlapping description will be appropriatelyomitted.

Embodiment

As methods of analyzing phenomenon of general material science using acomputer based on classical mechanics or quantum mechanics, simulationsbased on a molecular dynamics method (hereinafter, referred to as “MDmethod”), a quantum molecular dynamics method, or a renormalizedmolecular dynamics (hereinafter, referred to as “RMD method”) which isdeveloped version of the MD method to operate with a macroscale systemare known (for example, see Japanese Unexamined Patent ApplicationPublication No. 2009-37334). To be precise, the MD method or the RMDmethod is a kinetic method (calculation of a physical quantity usesstatistical mechanics), and the particle method is a method whichdiscretizes a differential equation describing a continuum. However, inthis specification, the MD method or the RMD method is referred to asthe particle method.

Since the particle method can analyze not only static phenomenon butalso dynamic phenomenon, such as flow, the particle method attractsattention as a simulation method replacing the above-described finiteelement method in which the object to be analyzed is primarily staticphenomenon.

From a differential point of view, the particle method obtains aparticle system to be analyzed by discretizing a continuum by particles.For example, when a fluid is analyzed in the particle method, it isoften the case that the Navier-Stokes equation is discretized byparticles.

From another or integral point of view, the particle method forms acontinuum by collecting many particles. This is, for example, a view inwhich a large iron ball is formed by collecting and solidifying smalliron particles.

In general, when calculating a magnetic field at a certain point in aspace where many magnetic moments exist, due to superposition principle,the magnetic field at the certain point created by each magnetic momentis summed up over the magnetic moments. The inventors have foundaffinity between the method of calculating the magnetic field from acollection of magnetic moments and the particle method in the integralview, and have devised the application of a magnetic moment to each ofparticles in the particle method. With this, it is possible to broadenthe cover range of the particle method to magnetic field analysis whilemaintaining the advantage of the particle method, the advantage that theparticle method is effective for analysis of convection or largedisplacement.

The inventors have devised that it is possible to broaden the coverrange of the particle method to dynamic magnetic field analysisincluding electromagnetic induction phenomenon by taking intoconsideration induced magnetization induced in each particle in additionto an interaction term between magnetic moments applied to the particlesin the particle method. With this, it is possible to realize dynamicmagnetic field analysis by the particle method for physical phenomenon,which is accompanied with large displacement and requires considerationconcerning electromagnetic induction, such as interaction between astator and a rotor in an induction motor.

FIG. 1 is a block diagram showing the functions and configuration of ananalyzer 100 according to an embodiment. It should be understood thatrespective blocks shown in the drawing can be realized by hardware, forexample, an element, such as a central processing unit (CPU) of acomputer, or a mechanical device, or can be realized by software, forexample, a computer program. In the drawing, functional blocks which arerealized by cooperation of hardware and software are shown. Therefore,it should be understood by those skilled in the art in contact with thisspecification that these functional blocks can be realized by acombination of hardware and software in various ways.

In this embodiment, although a case where a particle system is analyzedbased on the RMD method is described, it is obvious to those skilled inthe art in contact with this specification that the technical concept ofthis embodiment can be applied to a case where the particle system isanalyzed based other particle methods, such as an MD method withoutrenormalization, a distinct element method (DEM), smoothed particlehydrodynamics (SPH), or moving particle semi-implicit (MPS).

The analyzer 100 is connected to an input device 102 and a display 104.The input device 102 may be a keyboard, a mouse, or the like whichreceives an input of a user related to processing executed on theanalyzer 100. The input device 102 may be configured to receive an inputfrom a network, such as Internet, or a recording medium, such as a CD ora DVD.

The analyzer 100 includes a particle system acquisition unit 108, amagnetic moment application unit 110, a numerical calculation unit 120,a display control unit 118, and a particle data storage unit 114.

The particle system acquisition unit 108 acquires data of a particlesystem having N (where N is a natural number) particles defined in aone, two, or three-dimensional virtual space based on input informationacquired from the user through the input device 102. The particle systemis a particle system which is renormalized using the RMD method.

The particle system acquisition unit 108 arranges the N particles in thevirtual space based on the input information and applies a speed to eachof the arranged particles. The particle system acquisition unit 108registers a particle ID for specifying an arranged particle, theposition of the particle, and the speed of the particle in the particledata storage unit 114 in association with one another.

The magnetic moment application unit 110 applies a magnetic moment toeach of the particles of the particle system acquired by the particlesystem acquisition unit 108 based on the input information acquired fromthe user through the input device 102. For example, the magnetic momentapplication unit 110 requests the user to input the magnetic moment ofeach of the particles of the particle system through the display 104.The magnetic moment application unit 110 registers the input magneticmoment in the particle data storage unit 114 in association with theparticle ID.

The following description will be provided assuming that all particlesof the particle system are homogenous or equivalent and a potentialenergy function is based on pair potential and has the same formregardless of particles. However, it is obvious to those skilled in theart in contact with this specification that the technical concept ofthis embodiment can be applied to other cases.

The numerical calculation unit 120 numerically calculates a governingequation which governs the movement of each particle of the particlesystem represented by data stored in the particle data storage unit 114.In particular, the numerical calculation unit 120 performs repetitivecalculation according to an equation of motion of a discretizedparticle. The equation of motion of the particle has a term dependent onthe magnetic moment.

The numerical calculation unit 120 includes a magnetic field calculationunit 121, a force calculation unit 122, a particle state calculationunit 124, a state update unit 126, and an end condition determinationunit 128.

The magnetic field calculation unit 121 calculates a magnetic fieldgenerated by the particle system represented by data stored in theparticle data storage unit 114. The magnetic field calculation unit 121calculates the magnetic field based on the magnetic moment of eachparticle stored in the particle data storage unit 114. The magneticfield is a magnetic physical quantity relating to the particle system.The magnetic field calculation unit 121 may calculate magnetic fluxdensity or magnetization instead of or in addition to the magneticfield.

The force calculation unit 122 refers to data of the particle systemstored in the particle data storage unit 114 and calculates a forceapplied to the particle based on the inter-particle distance for eachparticle of the particle system. The force applied to the particleincludes a force based on interaction between the magnetic moments. Theforce calculation unit 122 determines particles (hereinafter, referredto as near particles) whose distance from an i-th (where 1≦i≦N) particleis less than a predetermined cut-off distance for the i-th particle ofthe particle system.

For each near particle, the force calculation unit 122 calculates aforce applied to the i-th particle by the near particle based on thepotential energy function between the near particle and the i-thparticle and the distance between the near particle and the i-thparticle. In particular, the force calculation unit 122 calculates theforce from the value of the gradient of the potential energy function atthe value of the distance between the near particle and the i-thparticle. The force calculation unit 122 sums up the force applied tothe i-th particle by the near particle for all near particles tocalculate the force applied to the i-th particle.

The particle state calculation unit 124 refers to data of the particlesystem stored in the particle data storage unit 114 and applies theforce calculated by the force calculation unit 122 to the equation ofmotion of the discretized particle for each particle of the particlesystem to calculate at least one of the position and speed of theparticle. In this embodiment, the particle state calculation unit 124calculates both the position and speed of the particle.

The particle state calculation unit 124 calculates the speed of theparticle from the equation of motion of the discretized particleincluding the force calculated by the force calculation unit 122. Theparticle state calculation unit 124 substitutes the force calculated bythe force calculation unit 122 in the equation of motion of thediscretized particle using a predetermined minute time interval Δt basedon a predetermined numerical analysis method, such as a leapfrog methodor a Euler method, for the i-th particle of the particle system, therebycalculating the speed of the particle. In the calculation, the speed ofthe particle calculated in the previous repetitive calculation cycle isused.

The particle state calculation unit 124 calculates the position of theparticle based on the calculated speed of the particle. The particlestate calculation unit 124 applies the calculated speed of the particleto a relationship expression of the position and speed of thediscretized particle using the time interval Δt based on a predeterminednumerical analysis method for the i-th particle of the particle system,thereby calculating the position of the particle. In the calculation,the position of the particle calculated in the previous repetitivecalculation cycle is used.

The state update unit 126 updates the position and speed of eachparticle of the particle system stored in the particle data storage unit114 to the position and speed calculated by the particle statecalculation unit 124.

The end condition determination unit 128 performs determination aboutwhether or not to end repetitive calculation in the numericalcalculation unit 120. An end condition for ending repetitive calculationis, for example, that repetitive calculation is performed apredetermined number times, an end instruction is received from theoutside, or the particle system reaches a steady state. When the endcondition is satisfied, the end condition determination unit 128 endsrepetitive calculation in the numerical calculation unit 120. When theend condition is not satisfied, the end condition determination unit 128returns the process to the force calculation unit 122. When thishappens, the force calculation unit 122 calculates the force with theposition and speed of the particle updated by the state update unit 126again.

The display control unit 118 causes the display 104 to display the formof a time expansion of the particle system or the state of the particlesystem at a certain time based on the position, speed, and magneticmoment of each particle of the particle system represented by datastored in the particle data storage unit 114. The display may beperformed in a form of a still image or a motion image.

FIG. 2 is a data structure diagram showing an example of the particledata storage unit 114. The particle data storage unit 114 stores theparticle ID, the position of the particle, the speed of the particle,and the magnetic moment of the particle in association with one another.

In the above-described embodiment, an example of the storage unit is ahard disk or a memory. It should be understood by those skilled in theart in contact with this specification that the respective units can berealized by a CPU (not shown), a module of an installed applicationprogram, a module of a system program, a memory which temporarily storesthe contents of data read from a hard disk, or the like based on thedescription of the specification.

The operation of the analyzer 100 having the above-describedconfiguration will be described.

FIG. 3 is a flowchart showing a sequence of processing in the analyzer100. The particle system acquisition unit 108 acquires a particle systemwhich is renormalized based on the RMD method (S12). The magnetic momentapplication unit 110 applies a magnetic moment to each of particles ofthe acquired particle system (S14). The magnetic field calculation unit121 calculates a magnetic field generated by the particle system (S16).The force calculation unit 122 calculates a force applied to a particlefrom the inter-particle distance and the magnetic moment of eachparticle (S18). The particle state calculation unit 124 calculates thespeed and position of a particle from the equation of motion of theparticle including the calculated force (S20). The state update unit 126updates the position and speed of the particle stored in the particledata storage unit 114 to the calculated position and speed (S22). Theend condition determination unit 128 performs determination aboutwhether or not the end condition is satisfied (S24). When the endcondition is not satisfied (N in S24), the process repeats Steps S16 toS22. When the end condition is satisfied (Y in S24), the display controlunit 118 outputs the calculation result (S26).

In general, the particle method can perform preferred analysis of adynamic object to be analyzed, such as a fluid, separation phenomenon,or cutting work.

Then, in the analyzer 100 of this embodiment, it is possible to realizethe analysis based on the particle method and the magnetic fieldanalysis based on the magnetic moment. Therefore, it is possible toanalyze a dynamic object to be analyzed with high accuracy and in ashort time, including a magnetic property, such as a magnetic field.

For example, when the object to be analyzed is iron sand, it isdifficult to simulate the object since preferred arrangement of mesh isnot possible with a related art mesh-based analysis method, such as thefinite element method. Therefore, in the analyzer 100 of thisembodiment, it is possible perform preferred analysis of iron sand withthe particle method in consideration of the magnetic property of ironsand.

For example, if the object to be analyzed is a motor, the related artmesh-based analysis method is not effective to the object since a rotorrotates with respect to a stator, and allows static analysis only.However, in the analyzer 100 of this embodiment, it is possible todynamically analyze magnetic interaction between the rotor and thestator while simulating the rotation of the rotor using the particlemethod with high accuracy. Furthermore, even if the object to beanalyzed includes electromagnetic induction phenomenon, such as aninduction motor, induced magnetization in consideration ofelectromagnetic induction phenomenon and a magnetic moment based on theinduced magnetization are applied, whereby it is possible to analyzemagnetic interaction with a time variation in a magnetic field.

Hereinafter, the principle of the analysis method used for the analyzer100 will be described.

The magnetic field can be obtained from a macroscopic vector potential

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 1} \right\rbrack & \; \\{{A(x)} = {\frac{\mu_{0}}{4\pi}{\int{\frac{J\left( x^{\prime} \right)}{{x - x^{\prime}}}{^{3}x^{\prime}}}}}} & \;\end{matrix}$

as follows. The direction of spin is defined as a z axis, and a isdefined as a radius of an atom. A circle current which generates spinS_(jz) (magnetic moment m_(jz)) is expressed as follows.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack & \; \\{{I_{j} = \frac{m_{jz}}{\pi \; a^{2}}},{m_{jz} = {g\; \mu_{B}S_{jz}}}} & \;\end{matrix}$

A magnetic field generated by a single spin at a lattice point j isexpressed as follows.

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack} & \; \\{\mspace{79mu} {{B_{r}\left( {x_{p} - x_{j}} \right)} = {\frac{\mu_{0}m_{jz}}{2\pi \; {ar}_{pj}}{\sum\limits_{n = 0}^{\infty}{\frac{\left( {- 1} \right)^{n}{\left( {{2n} + 1} \right)!!}}{2^{n}{n!}}\frac{r_{{pj} <}^{{2n} + 1}}{r_{{pj} >}^{{2n} + 2}}{P_{{2n} + 1}\left( {\cos \; \theta_{j}} \right)}}}}}} & \left( {P \cdot 1} \right) \\{{B_{\theta}\left( {x_{p} - x_{j}} \right)} = {{- \frac{\mu_{0}m_{jz}}{4\pi}}{\sum\limits_{n = 0}^{\infty}{\frac{\left( {- 1} \right)^{n}{\left( {{2n} + 1} \right)!!}}{2^{n}{\left( {n + 1} \right)!}}\begin{Bmatrix}{{- \left( \frac{{2n} + 2}{{2n} + 1} \right)}\frac{1}{a^{3}}\left( \frac{r_{pj}}{a} \right)^{2n}} \\{\frac{1}{r_{pj}^{3}}\left( \frac{a}{r_{pj}} \right)^{2n}}\end{Bmatrix}{P_{{2n} + 1}^{1}\left( {\cos \; \theta_{j}} \right)}}}}} & \left( {P \cdot 2} \right) \\{\mspace{79mu} {{B_{\varphi}\left( {x_{p} - x_{j}} \right)} = 0}} & \left( {P \cdot 3} \right)\end{matrix}$

Here, the following relationship is established.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack & \; \\{{r_{pj} = {{x_{p} - x_{j}}}},{{\cos \; \theta_{j}} = \frac{z_{p} - z_{j}}{r_{pj}}}} & \;\end{matrix}$

If only n=0 is left, the magnetic field becomes equivalent to themagnetic field created by a sphere which is uniformly magnetized.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 5} \right\rbrack & \; \\{M_{j} = {{m_{j}/\frac{4}{3}}\pi \; a^{3}}} & \;\end{matrix}$

When this method is applied to a bulk spherical body, if only n=0 isused, uniform magnetization may not be obtained, and the result will besimilar to the magnetic moment method. This may be improved by addinghigher-order terms. Uniform magnetization is obtained in the FEM.

The transformation to a Cartesian coordinate system (using the followingexpression) is performed.

x _(pj) =r _(pj) sin θ_(j) cos φ_(j) ,y _(pj) =r _(pj) sin θ_(j) sinφ_(j)  [Equation 6]

Then, the following expressions are defined.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 7} \right\rbrack & \; \\\begin{matrix}{{B_{x}\left( {{x_{p} - x_{j}};m_{jz}} \right)} = {\left( {{{B_{r}\left( m_{z} \right)}\sin \; \theta_{j}} + {{B_{\theta}\left( m_{z} \right)}\cos \; \theta_{j}}} \right)\cos \; \varphi_{j}}} \\{= {\left( {B_{r} + {B_{\theta}\cos \; \theta_{j}}} \right)\frac{x_{pj}}{r_{pj}}}}\end{matrix} & \left( {P \cdot 4} \right) \\\begin{matrix}{{B_{y}\left( {{x_{p} - x_{j}};m_{jz}} \right)} = {\left( {{{B_{r}\left( m_{z} \right)}\sin \; \theta_{j}} + {{B_{\theta}\left( m_{z} \right)}\cos \; \theta_{j}}} \right)\sin \; \varphi_{j}}} \\{= {\left( {B_{r} + {B_{\theta}\cos \; \theta_{j}}} \right)\frac{y_{pj}}{r_{pj}}}}\end{matrix} & \left( {P \cdot 5} \right) \\{{B_{z}\left( {{x_{p} - x_{j}};m_{jz}} \right)} = {{{B_{r}\left( m_{z} \right)}\cos \; \theta_{j}} - {{B_{\theta}\left( m_{z} \right)}\sin \; \theta_{j}}}} & \left( {P \cdot 6} \right)\end{matrix}$

Here, it should be noted that, when sin θ_(j)=0, B_(x)=B_(y)=0 fromB_(θ)=0. The magnetic fields when the magnetic moments m_(x) and m_(y)are respectively parallel to the x axis and the y axis are obtained in asimilar way.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 8} \right\rbrack & \; \\{{B\left( {;m_{x}} \right)},{B\left( {;m_{y}} \right)}} & \; \\\left\lbrack {{Equation}\mspace{14mu} 9} \right\rbrack & \; \\\begin{matrix}{{B_{y}\left( {{x_{p} - x_{j}};m_{jx}} \right)} = {\left( {{{B_{r}\left( m_{x} \right)}\sin \; \theta_{j}} + {{B_{\theta}\left( m_{x} \right)}\cos \; \theta_{j}}} \right)\cos \; \varphi_{j}}} \\{= {\left( {B_{r} + {B_{\theta}\cos \; \theta_{j}}} \right)\frac{y_{pj}}{r_{pj}}}}\end{matrix} & \left( {P \cdot 7} \right) \\\begin{matrix}{{B_{z}\left( {{x_{p} - x_{j}};m_{jx}} \right)} = {\left( {{{B_{r}\left( m_{x} \right)}\sin \; \theta_{j}} + {{B_{\theta}\left( m_{x} \right)}\cos \; \theta_{j}}} \right)\sin \; \varphi_{j}}} \\{= {\left( {B_{r} + {B_{\theta}\cos \; \theta_{j}}} \right)\frac{z_{pj}}{r_{pj}}}}\end{matrix} & \left( {P \cdot 8} \right) \\{{B_{x}\left( {{x_{p} - x_{j}};m_{jx}} \right)} = {{{B_{r}\left( m_{x} \right)}\cos \; \theta_{j}} - {{B_{\theta}\left( m_{x} \right)}\sin \; \theta_{j}}}} & \left( {P \cdot 9} \right) \\\begin{matrix}{{B_{z}\left( {{x_{p} - x_{j}};m_{jy}} \right)} = {\left( {{{B_{r}\left( m_{y} \right)}\sin \; \theta_{j}} + {{B_{\theta}\left( m_{y} \right)}\cos \; \theta_{j}}} \right)\cos \; \varphi_{j}}} \\{= {\left( {B_{r} + {B_{\theta}\cos \; \theta_{j}}} \right)\frac{z_{pj}}{r_{pj}}}}\end{matrix} & \left( {P \cdot 10} \right) \\\begin{matrix}{{B_{x}\left( {{x_{p} - x_{j}};m_{jy}} \right)} = {\left( {{{B_{r}\left( m_{y} \right)}\sin \; \theta_{j}} + {{B_{\theta}\left( m_{y} \right)}\cos \; \theta_{j}}} \right)\sin \; \varphi_{j}}} \\{= {\left( {B_{r} + {B_{\theta}\cos \; \theta_{j}}} \right)\frac{x_{pj}}{r_{pj}}}}\end{matrix} & \left( {P \cdot 11} \right) \\{{B_{y}\left( {{x_{p} - x_{j}};m_{jy}} \right)} = {{{B_{r}\left( m_{y} \right)}\cos \; \theta_{j}} - {{B_{\theta}\left( m_{y} \right)}\sin \; \theta_{j}}}} & \left( {P \cdot 12} \right)\end{matrix}$

The Magnetic Field

B _(m)  [Equation 11]

at

x _(p)  [Equation 10]

(contributions to N spins) is as follows.

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 12} \right\rbrack} & \; \\{{B_{m}\left( x_{p} \right)} = {\sum\limits_{j \neq p}^{N}\left\lbrack {{B\left( {{x_{p} - x_{j}};m_{jx}} \right)} + {B\left( {{x_{p} - x_{j}};m_{jy}} \right)} + {B\left( {{x_{p} - x_{j}};m_{jz}} \right)}} \right\rbrack}} & \left( {P{.13}} \right) \\{\mspace{79mu} {M_{p} = {\frac{3}{\mu_{0}}\left( \frac{\mu - \mu_{0}}{\mu + {2\mu_{0}}} \right){B_{o}\left( x_{p} \right)}}}} & \left( {P{.14}} \right) \\{\mspace{79mu} {{B_{o}\left( x_{p} \right)} = {{B_{m}\left( x_{p} \right)} + {\mu_{0}H_{ext}}}}} & \left( {P{.15}} \right) \\{\mspace{79mu} {m_{p} = {\left( {m_{jx},m_{jy},m_{jz}} \right) = {\frac{4}{3}\pi \; a^{3}M_{p}}}}} & \left( {P{.16}} \right)\end{matrix}$

Here, the following expression means the total external field.

B _(o)  [Equation 13]

By combining P-14 and P-15, the magnetic moment m_(p) is obtained. Themagnetic field generated outside the magnetic body is expressed by P-15,and the magnetic field generated inside the magnetic body is expressedby the following expression.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 14} \right\rbrack & \; \\{B_{in} = {B_{o} + {\frac{2\mu_{o}}{3}M}}} & \left( {P \cdot 17} \right) \\{H_{in} = {{\frac{1}{\mu_{o}}B_{o}} - {\frac{1}{3}M}}} & \left( {P \cdot 18} \right)\end{matrix}$

P-14 satisfies the following expression in consideration of ademagnetizing field due to magnetization generated on the surface of thesphere.

∇·M _(p)=0  [Equation 15]

P-13 is slow to converge since the matrix is not a superdiagonal angle(j=p is excluded). Thus, a solution is obtained from the followingequation of motion.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 16} \right\rbrack & \; \\{\mathcal{L} = {\sum\limits_{i}\left\lbrack {{\frac{1}{2}{m_{v}\left( {1 + \chi} \right)}\left( {{\overset{.}{m}}_{i} \cdot {\overset{.}{m}}_{i}} \right)} - {\frac{1}{2}\left( {m_{i} - {\alpha \; {B_{o}\left( r_{i} \right)}}} \right)^{2}}} \right\rbrack}} & \left( {P \cdot 19} \right) \\{{m_{v}\frac{{\overset{.}{m}}_{i}}{t}} = {{{- \frac{1}{1 + \chi}}\left( {m_{i} - {\alpha \; {B_{o}\left( r_{i} \right)}}} \right)} - {\gamma \; {\overset{.}{m}}_{i}}}} & \left( {P \cdot 20} \right) \\{\alpha = {\frac{4\pi \; a^{3}}{\mu_{o}}\left( \frac{\mu - \mu_{o}}{\mu + {2\mu_{o}}} \right)}} & \left( {P \cdot 21} \right)\end{matrix}$

Here, m_(v)=5×10⁻²dt² is a virtual mass and γ=m_(v)/(10 dt) is a dampingcoefficient.

It is possible to realize further acceleration with an addition of FIRE(see Erik Bitzek et al., “Structural Relaxation Made Simple”, PhysicalReview Letters, Oct. 27, 2006, 97). The code of the FIRE is shown below.The calculation result is 10.05 [s] under the condition that the numberof particles is 802 and the residual is <10⁻⁸. The residual is not below10⁻⁵ without the FIRE.

Ordinary Step

Calculate m, force, and {dot over (m)} from Eq.(P•20)

Fire Step

P=force·{dot over (m)};

{dot over (m)} _(a) =|{dot over (m)}|,f _(a)=|force|=DLB_MIN;

{dot over (m)}={dot over (m)}(1.0−α)+(force/f _(a)){dot over (m)}_(a)α;  [Equation 17]

if (P>0.0) n++;if (n>5){α=0.99α; n=0;}if (P<=0.0){m=0.0; a=0.1; n=0;}

Special cases of the Legendre functions and the associated Legendrefunction are listed below (x: =cos θ_(j))

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 18} \right\rbrack & \; \\{{P_{0}(x)} = 1} & \left( {P{.22}} \right) \\{{P_{1}(x)} = x} & \left( {P{.23}} \right) \\{{P_{2}(x)} = {\frac{1}{2}\left( {{3\; x^{2}} - 1} \right)}} & \left( {P{.24}} \right) \\{{P_{3}(x)} = {\frac{1}{2}\left( {{5\; x^{2}} - {3\; x}} \right)}} & \left( {P{.25}} \right) \\{{P_{4}(x)} = {\frac{1}{8}\left( {{35\; x^{4}} - {30\; x^{2}} + 3} \right)}} & \left( {P{.26}} \right) \\{{P_{5}(x)} = {\frac{1}{8}\left( {{63\; x^{5}} - {70\; x^{3}} + {15\; x}} \right)}} & \left( {P{.27}} \right) \\{and} & \left( {P{.28}} \right) \\{{P_{1}^{1}(x)} = {- \left( {1 - x^{2}} \right)^{1/2}}} & \left( {P{.29}} \right) \\{{P_{3}^{1}(x)} = {{- \frac{3}{2}}\left( {1 - x^{2}} \right)^{1/2}\left( {{5\; x^{2}} - 1} \right)}} & \left( {P{.30}} \right) \\{{P_{3}^{2}(x)} = {15\left( {1 - x^{2}} \right)x}} & \left( {P{.31}} \right) \\{{P_{3}^{3}(x)} = {{- 15}\left( {1 - x^{2}} \right)^{3/2}}} & \left( {P{.32}} \right) \\{{P_{5}^{1}(x)} = {{- \frac{1}{8}}\left( {1 - x^{2}} \right)^{1/2}\left( {{315\; x^{4}} - {210\; x^{2}} + 15} \right)}} & \left( {P{.33}} \right)\end{matrix}$

Hereinafter, the magnetization

M  [Equation 19]

of a magnetic sphere (the magnetic susceptibility is 999) in a uniformexternal field of B_(ox)=0.1 [T] in the x-axis direction is calculated,and the result of the calculation is shown below.

The number of particles (atoms) forming the sphere is 4009 and theparticles are arranged in an fcc structure. The exact solution isexpressed as follows.

$\begin{matrix}{M = {\frac{3}{\mu_{o}}\left( \frac{\mu - \mu_{o}}{\mu + {2\; \mu_{o}}} \right)B_{o}}} & \left\lbrack {{Equation}\mspace{14mu} 20} \right\rbrack\end{matrix}$

FIGS. 4, 5, 6, and 7 show the results of the first to fourth orders ofmultipole expansion. Cross-sections which pass the center are shown. Ifthe order of the multipole expansion increases, it can be seen that theresults get closer to the exact solution. If the order is low, theconvergence is deteriorated. Most of the calculation time is occupiedwith calls for spherical harmonics.

FIG. 4 is a diagram showing the calculation results corresponding to anexpansion to n=1 order. The calculation value is 0.346231 [T] on averagewith respect to the exact solution μ_(o)M_(x)=0.2991 [T]. It can be seenthat the magnetization is uniform and parallel to the external fieldB_(x)=0.1 [T]. The residual is equal to or less than 1e-13 and thecalculation time is about 80 minutes.

FIG. 5 is a diagram showing the calculation results corresponding to anexpansion to n=2 order. The calculation value is 0.321891 [T] on averagewith respect to the exact solution μ_(o)M_(x)=0.2991 [T]. It can be seenthat the magnetization is uniform and parallel to the external fieldB_(x)=0.1 [T]. The residual is equal to or less than 1e-8 and thecalculation time is about 3 hours. The residual does not fall below1e-9.

FIG. 6 is a diagram showing calculation results corresponding to anexpansion to n=3 order. The calculation value is 0.312444 [T] on averagewith respect to the exact solution μ_(o)M_(x)=0.2991 [T]. It can be seenthat the magnetization is uniform and parallel to the external fieldB_(x)=0.1 [T].

FIG. 7 is a diagram showing calculation results corresponding to anexpansion to n=4 order. The calculation value is 0.312222 [T] on averagewith respect to the exact solution μ_(o)M_(x)=0.2991 [T]. It can be seenthat the magnetization is uniform and parallel to the external fieldB_(x)=0.1 [T]. The residual does not fall below 1e-9 after about 5hours.

FIG. 8 is a diagram showing the relationship between the orders ofspherical harmonics and calculation values. It can be seen that a higherorder gives results closer to the exact solution.

The principle of the analysis method used for the analyzer 100 may bedescribed as follows.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 21} \right\rbrack & \; \\{L = {L_{MD} + L_{ED} + L_{EM}}} & \left( {Q\text{-}1} \right) \\{L_{MD} = {\sum\limits_{p}^{N}\; \left\lbrack {{\frac{1}{2}M_{p}V_{p}^{2}} - {\Phi \left( x_{p} \right)} - {{{eA}\left( x_{p} \right)} \cdot V_{p}}} \right\rbrack}} & \left( {Q\text{-}2} \right) \\{L_{ED} = {\sum\limits_{e}^{N}\; \left\lbrack {{\frac{1}{2}m_{e}v_{e}^{2}} + {{{eA}\left( x_{e} \right)} \cdot v_{e}}} \right\rbrack}} & \left( {Q\text{-}3} \right) \\{L_{EM} = {\sum\limits_{m}^{N}{\delta \; {V_{m}\left\lbrack {{\frac{\mu_{0}}{2}{m_{v}\left( {{\overset{.}{H}}_{om} \cdot {\overset{.}{H}}_{om}} \right)}} - {\frac{\mu_{0}}{2}{\lambda \left( {H_{om} - {H_{o}\left( x_{m} \right)}} \right)}^{2}}} \right\rbrack}}}} & \left( {Q\text{-}4} \right)\end{matrix}$

M_(p) is a mass of a nucleus, and m_(e) is amass of an electron. e is(an absolute value of) a charge of a nucleus or an electron. The chargeand the mass of an electron may not explicitly appear since the chargeis converted to a current and the mass of an electron is added to themass of a nucleus to give a mass M of an atom in the end. δV is a volumeof the magnetic sphere, m_(v) is a virtual mass, and λ_(i) is aLagrangian undetermined coefficient. To satisfy constraints rigorously,it is necessary to use convergence calculation like the SHAKE method;however, in this method, it is assumed that λ_(i)=1 and small vibrationsaround the true solution are treated as errors.

The direction of spin is defined as a z axis, and a is a radius of anatom. A circle current which generates spin S_(jz) (magnetic momentm_(jz)) at a lattice point j is expressed as follows.

$\begin{matrix}{{I_{j} = \frac{m_{jz}}{\pi \; a^{2}}},{m_{jz} = {g\; \mu_{B}S_{jz}}}} & \left\lbrack {{Equation}\mspace{14mu} 22} \right\rbrack\end{matrix}$

Therefore, the magnetic field can be obtained from the macroscopicvector potential

$\begin{matrix}{{A(x)} = {\frac{\mu_{0}}{4\; \pi}{\int{\frac{J\left( x^{\prime} \right)}{{x - x^{\prime}}}{^{3}x^{\prime}}}}}} & \left\lbrack {{Equation}\mspace{14mu} 23} \right\rbrack\end{matrix}$

as follows.

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 24} \right\rbrack} & \; \\{\mspace{79mu} {{H_{r}\left( r_{pj} \right)} = {\frac{m_{jz}}{2\; \pi \; {ar}_{pj}}{\sum\limits_{n = 0}^{\infty}{\frac{\left( {- 1} \right)^{n}{\left( {{2\; n} + 1} \right)!!}}{2^{n}{n!}}\frac{r_{{pj} <}^{{2\; n} + 1}}{r_{{pj} >}^{{2\; n} + 2}}{P_{{2\; n} + 1}\left( {\cos \; \theta_{j}} \right)}}}}}} & \left( {Q{.5}} \right) \\{{H_{\theta}\left( r_{pj} \right)} = {{- \frac{m_{jz}}{4\; \pi}}{\sum\limits_{n = 0}^{\infty}\; {\frac{\left( {- 1} \right){{n\left( {{2\; n} + 1} \right)}!!}}{2{{n\left( {n + 1} \right)}!}}\begin{Bmatrix}{{- \left( \frac{{2\; n} + 2}{{2\; n} + 1} \right)}\frac{1}{a^{3}}\left( \frac{r_{pj}}{a} \right)^{2\; n}} \\{\frac{1}{r_{pj}^{3}}\left( \frac{a}{r_{pj}} \right)^{2\; n}}\end{Bmatrix}{P_{{2\; n} + 1}^{1}\left( {\cos \; \theta_{j}} \right)}}}}} & \left( {Q{.6}} \right) \\{\mspace{79mu} {{H_{\varphi}\left( r_{pj} \right)} = 0}} & \left( {Q{.7}} \right)\end{matrix}$

Here, the following relationship is established.

$\begin{matrix}{{r_{pj} = {{x_{p} - x_{j}}}},{{\cos \; \theta_{j}} = \frac{z_{p} - z_{j}}{r_{pj}}}} & \left\lbrack {{Equation}\mspace{14mu} 25} \right\rbrack\end{matrix}$

If only n=0 is left, the magnetic field becomes equivalent to themagnetic field created by a sphere which is uniformly magnetized.

$\begin{matrix}{M_{j} = {{m_{j}/\frac{4}{3}}\pi \; a^{3}}} & \left\lbrack {{Equation}\mspace{14mu} 26} \right\rbrack\end{matrix}$

The transformation to a Cartesian coordinate system (using the followingexpression) is performed.

x _(pj) =r _(pj) sin θ_(j) cos φ_(j) ,y _(pi) =r _(pj) sin θ_(j) sinφ_(j)  [Equation 27]

Then, the following expressions are defined.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 28} \right\rbrack & \; \\\begin{matrix}{{H_{x}\left( {r_{pj};m_{jz}} \right)} = {\left\{ {{{H_{r}\left( {r_{pj};m_{z}} \right)}\sin \; \theta_{j}} + {{H_{\theta}\left( {r_{pj};m_{z}} \right)}\cos \; \theta_{j}}} \right\} \cos \; \varphi_{j}}} \\{= {\left( {H_{r} + {H_{\theta}\cot \; \theta_{j}}} \right)\frac{x_{pj}}{r_{pj}}}}\end{matrix} & \left( {Q{.8}} \right) \\\begin{matrix}{{H_{y}\left( {r_{pj};m_{jz}} \right)} = {\left\{ {{{H_{r}\left( {r_{pj};m_{z}} \right)}\sin \; \theta_{j}} + {{H_{\theta}\left( {r_{pj};m_{z}} \right)}\cos \; \theta_{j}}} \right\} \sin \; \varphi_{j}}} \\{= {\left( {H_{r} + {H_{\theta}\cot \; \theta_{j}}} \right)\frac{y_{pj}}{r_{pj}}}}\end{matrix} & \left( {Q{.9}} \right) \\{{H_{z}\left( {r_{pj};m_{jz}} \right)} = {{{H_{r}\left( {r_{pj};m_{z}} \right)}\cos \; \theta_{j}} - {{H_{\theta}\left( {r_{pj};m_{z}} \right)}\sin \; \theta_{j}}}} & \left( {Q{.10}} \right)\end{matrix}$

Here, it should be noted that, when sin θ_(j)=0, H_(x)=H_(y)=0 fromH_(θ)=0. The magnetic fields

H(;m _(x)),H(;m _(y))  [Equation 29]

when the magnetic moments m_(x) and m_(y) are respectively parallel tothe x axis and the y axis are obtained in a similar way.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 30} \right\rbrack & \; \\\begin{matrix}{{H_{y}\left( {r_{pj};m_{jx}} \right)} = {\left\{ {{{H_{r}\left( {r_{pj};m_{x}} \right)}\sin \; \theta_{j}} + {{H_{\theta}\left( {r_{pj};m_{x}} \right)}\cos \; \theta_{j}}} \right\} \cos \; \varphi_{j}}} \\{= {\left( {H_{r} + {H_{\theta}\cot \; \theta_{j}}} \right)\frac{y_{pj}}{r_{pj}}}}\end{matrix} & \left( {Q{.11}} \right) \\\begin{matrix}{{H_{z}\left( {r_{pj};m_{jx}} \right)} = {\left\{ {{{H_{r}\left( {r_{pj};m_{x}} \right)}\sin \; \theta_{j}} + {{H_{\theta}\left( {r_{pj};m_{x}} \right)}\cos \; \theta_{j}}} \right\} \sin \; \varphi_{j}}} \\{= {\left( {H_{r} + {H_{\theta}\cot \; \theta_{j}}} \right)\frac{z_{pj}}{r_{pj}}}}\end{matrix} & \left( {Q{.12}} \right) \\{{H_{x}\left( {r_{pj};m_{jx}} \right)} = {{{H_{r}\left( {r_{pj};m_{x}} \right)}\cos \; \theta_{j}} - {{H_{\theta}\left( {r_{pj};m_{x}} \right)}\sin \; \theta_{j}}}} & \left( {Q{.13}} \right) \\{{\cos \; \theta_{j}} = \frac{x_{p} - x_{j}}{r_{pj}}} & \left( {Q{.14}} \right) \\\begin{matrix}{{H_{z}\left( {r_{pj};m_{jy}} \right)} = {\left\{ {{{H_{r}\left( {r_{pj};m_{y}} \right)}\sin \; \theta_{j}} + {{H_{\theta}\left( {r_{pj};m_{y}} \right)}\cos \; \theta_{j}}} \right\} \cos \; \varphi_{j}}} \\{= {\left( {H_{r} + {H_{\theta}\cot \; \theta_{j}}} \right)\frac{z_{pj}}{r_{pj}}}}\end{matrix} & \left( {Q{.15}} \right) \\\begin{matrix}{{H_{x}\left( {r_{pj};m_{jy}} \right)} = {\left\{ {{{H_{r}\left( {r_{pj};m_{y}} \right)}\sin \; \theta_{j}} + {{H_{\theta}\left( {r_{pj};m_{y}} \right)}\cos \; \theta_{j}}} \right\} \sin \; \varphi_{j}}} \\{= {\left( {H_{r} + {H_{\theta}\cot \; \theta_{j}}} \right)\frac{x_{pj}}{r_{pj}}}}\end{matrix} & \left( {Q{.16}} \right) \\{{H_{y}\left( {r_{pj};m_{jy}} \right)} = {{{H_{r}\left( {r_{pj};m_{y}} \right)}\cos \; \theta_{j}} - {{H_{\theta}\left( {r_{pj};m_{y}} \right)}\sin \; \theta_{j}}}} & \left( {Q{.17}} \right) \\{{\cos \; \theta_{j}} = \frac{y_{p} - y_{j}}{r_{pj}}} & \left( {Q{.18}} \right)\end{matrix}$

The external magnetic field (contributions from N spins)

H _(m)  [Equation 32]

felt by a moment at

X _(p)  [Equation 31]

is expressed as follows.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 33} \right\rbrack & \; \\{{H_{m}\left( x_{p} \right)} = {\sum\limits_{j \neq p}^{N}\; \left\lbrack {{H\left( {r_{pj};m_{jx}} \right)} + {H\left( {r_{pj};m_{jy}} \right)} + {H\left( {r_{pj};m_{jz}} \right)}} \right\rbrack}} & \left( {Q{.19}} \right) \\{m_{p} = {\left( {m_{px},m_{py},m_{pz}} \right) = {M_{p}\delta \; V_{p}}}} & \left( {Q{.20}} \right)\end{matrix}$

The Induction Magnetic Field

H _(ind)  [Equation 34]

is derived from the following expression when a p point is inside aconductor

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 35} \right\rbrack & \; \\{{H_{ind}^{(i)}\left( x_{p} \right)} = {{M_{ant}\left( x_{p} \right)} + {\frac{1}{4\; \pi}{\sum\limits_{j}^{\;}\; \frac{{3\; {n\left( {n \cdot m_{ant}^{j}} \right)}} - m_{ant}^{j}}{x_{pj}^{3}}}}}} & \left( {Q{.21}} \right) \\{\; {{M_{ant}\left( x_{p} \right)} = {{- C_{1}}{\sum\limits_{j}^{\;}\; {\sigma_{j}{G\left( x_{pj} \right)}\frac{{DB}_{j}}{Dt}}}}}} & \left( {Q{.21}\text{-}2} \right) \\{{G\left( x_{pj} \right)} = \left\{ \begin{matrix}{\frac{1}{3}\frac{a_{j}^{3}}{x_{pj}}} & {x_{pj} > a_{j}} \\{\frac{a_{j}^{2}}{2} - \frac{x_{pj}^{2}}{6}} & {x_{pj} < a_{j}}\end{matrix} \right.} & \left( {Q{.21}\text{-}3} \right) \\{m_{ant}^{j} = {C_{2}{M_{ant}\left( x_{j} \right)}\delta \; V_{j}}} & \left( {Q{.21}\text{-}4} \right)\end{matrix}$

and is derived from the following expression when the p point is outsidethe conductor.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 36} \right\rbrack & \; \\{{H_{ind}^{(e)}\left( x_{p} \right)} = {\frac{1}{4\; \pi}{\sum\limits_{j}^{\;}\; \frac{{3\; {n\left( {n{\cdot m_{ant}^{j}}} \right)}} - m_{ant}^{j}}{x_{pj}^{3}}}}} & \left( {Q{.22}} \right)\end{matrix}$

C₁ and C₂ are compensation coefficients when compared to a limitexpressed as follows in, for example, the expression described in L. D.Landau, E. M. Lifshitz, L. P. Litaevskii, “Electrodynamics of ContinuousMedia 2nd ed”, Jan. 1, 1984, page 205

$\begin{matrix}{\sqrt{\frac{2}{\mu_{0}\sigma \; \omega}}\operatorname{>>}a} & \left\lbrack {{Equation}\mspace{14mu} 37} \right\rbrack\end{matrix}$

and take the values of C₁=⅓ and C₂=⅗.

The first term (or Q-21-2) on the right side of Q-21 is a term whichrepresents induced magnetization M_(ant) induced in each particle by atime-varying external magnetic field. The second term on the right sideof Q-21 and the right side of Q-22 are terms which represent a magneticfield obtained by interaction between magnetic moments m_(ant) based onthe induced magnetization of each particle represented by ExpressionQ-21-4. The induction magnetic field is represented in this way, wherebyit is possible to realize dynamic magnetic field analysis inconsideration of the influence of the time-varying external magneticfield.

When the renormalization is performed, the electrical conductivity σ isas follows.

σ′=σα^(δ/2+1)  [Equation 38]

Here, δ=2.

The Lagrange differential on the right side of Q-21-2

$\begin{matrix}\frac{{DB}_{j}}{Dt} & \left\lbrack {{Equation}\mspace{14mu} 39} \right\rbrack\end{matrix}$

can be replaced with

{dot over (B)} _(j)  [Equation 40]

since it is the particle method. It should be noted that transitionalmovements of magnetization can be considered by movements of particles;however, the rotations of the magnetization vectors should be consideredseparately (Landau and Lifshitz, Electromagnetism, Chapter 51).

The magnetic field created inside the magnetic body is as follows.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 41} \right\rbrack & \; \\{{B\left( x_{p} \right)} = {\mu_{0}\left\{ {{H_{o}\left( x_{p} \right)} + {\frac{2}{3}{M(H)}}} \right\}}} & ({Q23}) \\{{H\left( x_{p} \right)} = {{H_{o}\left( x_{p} \right)} - {\frac{1}{3}{M(H)}}}} & ({Q24}) \\{{\overset{.}{B}\left( x_{p} \right)} = {\mu_{0}\frac{1 + {\chi (H)}}{1 + {{\chi (H)}/3}}{{\overset{.}{H}}_{o}\left( x_{p} \right)}}} & ({Q25})\end{matrix}$

The relationship between the total external field H_(o)(x_(p)) and themagnetization M(H) is as follows.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 42} \right\rbrack & \; \\{{H_{o}\left( x_{p} \right)} = {{H_{m}\left( x_{p} \right)} + {H_{ext}\left( x_{p} \right)} + {H_{ind}^{(i)}\left( x_{p} \right)}}} & ({Q26}) \\{{M(H)} = \left\{ \begin{matrix}{f(H)} & {{NONLINEAR}\mspace{14mu} {MATERIAL}} \\{3\left( \frac{\mu - \mu_{0}}{\mu + {2\; \mu_{0}}} \right){H_{o}\left( x_{p} \right)}} & {{LINEAR}\mspace{14mu} {MATERIAL}}\end{matrix} \right.} & ({Q27})\end{matrix}$

By combining Q-26 and Q-27, the magnetic moment

m _(p)  [Equation 43]

is obtained.

The magnetic field outside the magnetic body is as follows. [Equation44]

B(x _(p))=μ₀ H _(o)(x _(p))  (Q27-1)

H(x _(p))=H _(o)(x _(p))  (Q27-2)

{dot over (B)}(x _(p))=μ₀ {dot over (H)} _(o)(x _(p))  (Q27-3)

H _(o)(x _(p))=H _(m)(x _(p))+H _(ext)(x _(p))+H _(ind) ^((e))(x_(p))  (Q27-4)

Nonlinear Magnetization and Hysteresis The Magnetic Field

H  [Equation 45]

is removed from Q-27 and Q-24.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 46} \right\rbrack & \; \\{{M = {f\left( {H_{o} - {\frac{1}{3}M}} \right)}},{{for}\mspace{14mu} {only}\mspace{14mu} {sphere}}} & \left( {Q \cdot 28} \right)\end{matrix}$

By solving Q-28, the following expression is defined (FIG. 9 shows thecase where B=f(H)).

M=g(H _(o))  [Equation 47]

FIG. 9 is a diagram showing a hysteresis curve. M is obtained from apoint of intersection of the hysteresis curve and B+2μ_(o)H=3B_(o). Now,the permanent magnetization is expressed as follows.

M _(c)  [Equation 48]

The B-H curve with the above permanent magnetization is given asfollows.

B=μ _(o)(M _(c) +αH),α<1.

This can be combined with

B+2μ_(o) H=μ _(o) H _(o)  [Equation 50]

to obtain the following expression.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 51} \right\rbrack & \; \\{M = {\frac{3}{2 + a}\left\{ {M_{c} - {\left( {1 - a} \right)H_{o}}} \right\}}} & \;\end{matrix}$

With regard to a general B=f(H), the Newton-Laphson method is used.

Particulation Method of Magnetic Field

Q-19 is slow to converge since the matrix is not a superdiagonal angle(j=p is excluded). Thus, a solution is obtained from the followingequation of motion.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 52} \right\rbrack & \; \\{\mathcal{L}_{EM} = {\sum\limits_{i}\; \left\lbrack {{\frac{\mu_{o}}{2}{m_{v}\left( {{\overset{.}{H}}_{oi} \cdot {\overset{.}{H}}_{oi}} \right)}} - {\frac{\mu_{o}}{2}\left\{ {H_{oi} - {H_{o}\left( x_{i} \right)}} \right\}^{2}}} \right\rbrack}} & \left( {Q \cdot 29} \right) \\{{m_{v}\frac{{\overset{.}{H}}_{oi}}{t}} = {- \left\lbrack {H_{oi} - {H_{o}\left( x_{p} \right)}} \right\rbrack}} & \left( {Q \cdot 30} \right)\end{matrix}$

Here, H_(O)(x_(p)) is represented by Q-26. Furthermore, m_(v) is avirtual mass. The recommended value is 250×dt×dt. A sufficiently smallm_(v) can induce minute attenuation vibration around the true solution,whereby errors can be reduced.

A model is considered.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 53} \right\rbrack & \; \\{{m_{v}\frac{\overset{.}{H}}{t}} = {{{- k}\left\{ {H - {H_{o}{\cos ({wt})}}} \right\}} - {\gamma \; \overset{.}{H}}}} & \;\end{matrix}$

A solution of forced vibration having an attenuation term (Landau andLifshitz, Mechanics, Sec. 26) is as follows.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 54} \right\rbrack & \; \\{{H = {{a\; ^{{- \lambda}\; t}{\cos \left( {{\omega_{c}t} + \alpha} \right)}} + {b\; {\cos \left( {{\omega \; t} + \delta} \right)}}}},} & \left( {Q \cdot 32} \right) \\{{\lambda = \frac{\gamma}{2\; m}},{\omega_{o} = \sqrt{\frac{k}{m_{v}}}},{\omega_{c} = \sqrt{\omega_{o}^{2} - \lambda^{2}}},} & \left( {Q \cdot 33} \right) \\{{b = \frac{{kH}_{o}/m_{v}}{\sqrt{\left( {\omega_{o}^{2} - \omega^{2}} \right)^{2} + {4\; \lambda^{2}\omega^{2}}}}},{{\tan \; \delta} = \frac{2\; \lambda \; \omega}{\omega^{2} - \omega_{o}^{2}}}} & \left( {Q \cdot 34} \right)\end{matrix}$

If the following condition is considered,

ω<<ω_(o),ω<<λ  [Equation 55]

the following expressions are defined.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 56} \right\rbrack & \; \\{{H = {b\; {\cos \left( {{wt} + \delta} \right)}}},} & \left( {Q \cdot 35} \right) \\{{b = \frac{{kH}_{o}}{\sqrt{k^{2} + {\gamma^{2}\omega^{2}}}}},{{\tan \; \delta} = {{- \frac{\gamma}{k}}\omega}}} & \left( {Q \cdot 36} \right)\end{matrix}$

That is, if m_(v) is sufficiently small, the solution matches thesolution of the following essential equation to be solved.

γH=−k{H−H _(o) cos(ωt)}  [Equation 57]

Discretization

If the discretization is performed according to the leapfrog method (thespeed is evaluated at an intermediate point between n and n+1), thefollowing expressions are defined where the number of convergences is s.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 58} \right\rbrack & \; \\{{m_{v}\frac{{\overset{.}{H}}_{oi}^{n + {1/2}} - {\overset{.}{H}}_{oi}^{n - {1/2}}}{\delta \; t}} = {- \left\lbrack {H_{oi}^{n} - {H_{o}^{n}\left( x_{i} \right)}} \right\rbrack}} & \left( {{Q \cdot 37}\text{-}1} \right) \\{{{\overset{.}{H}}_{oi}^{s + 1} = {{\overset{.}{H}}_{oi}^{s} + {err}}},} & \left( {{Q \cdot 37}\text{-}2} \right) \\{{err} = {{- {\overset{.}{H}}_{oi}^{s}} + {\overset{.}{H}}_{oi}^{n - {1/2}} - {\frac{\delta \; t}{m_{v}}\left\lbrack {H_{oi}^{n} - {H_{o}^{n}\left( x_{i} \right)}} \right\rbrack}}} & \left( {{Q \cdot 37}\text{-}3} \right)\end{matrix}$

Here, in the discretization of the induction magnetic fieldH_(ind)(x_(i)), the following expression is defined.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 59} \right\rbrack & \; \\{{\overset{.}{B}}^{n} = {\frac{\mu_{0}}{2}\left( {{\overset{.}{H}}_{o}^{s} + {\overset{.}{H}}_{o}^{n - {1/2}}} \right)}} & \left( {Q \cdot 38} \right)\end{matrix}$

In this case, if only diagonal elements use further values, an explicitsolution is possible and the efficiency will be improved by about 30%.Most of the calculation is calls for special functions.

Acceleration Based on FIRE: Effective Only to Magnetostatic FieldAnalysis

It is possible to realize further acceleration with an addition of FIRE(see Erik Bitzek et al., “Structural Relaxation Made Simple”, PhysicalReview Letters, Oct. 27, 2006, 97). The code of the FIRE is shown below.The calculation result is 10.05 [s] under the condition that the numberof particles is 802 and the residual is <10⁻⁸. The residual does notfall below 10⁻⁵ without the FIRE.

-   -   Ordinary Step (Q•39)        Calculate H, force, and {dot over (H)} fromEq. (Q•30) (Q•40)    -   Fire Step (Q•41)

P=force·{dot over (H)};

{dot over (m)} _(a) =|{dot over (H)}|,f _(a)=|force|+DBL_MIN;

{dot over (H)}={dot over (H)}(1.0−α)+(force/f _(a)){dot over (H)}_(a)α;  [Equation 60]

if (P>0.0) n++;if (n>5){α=0.99α;n=0;}if (P<=0.0){{dot over (H)}=0.0;α=0.1;n=0;}

Accuracy Verification: Uniform Magnetization of Sphere

FIG. 10 is a diagram showing an analysis model of a conductive spherehaving beads. In the accuracy verification, a uniform external magneticfield was applied in a space to be H_(ext,x)=H_(O) sin (2πft) in the xaxis direction. Here, H_(O)=7.958×10⁴ [A/m], f=200 [Hz], the radius ofthe conductive sphere A=10 [mm], and electrical conductivity π=59×10⁶[S/m]. The number of particles (atoms) forming the conductive sphere was6099 and the particles were arranged in an fcc structure.

FIG. 11 is a graph showing calculation values of a time-varying magneticfield on a conductive sphere surface, and shows a time variation in amagnetic field Hx at the gravity center position (rx=9.89 [mm], ry=rz=0)of a particle near a conductor surface. In this drawing, the externalmagnetic field and the value by the exact solution are shown along withcalculation results by a simulation. It is possible to reproduce a formin which the magnetic field inside the conductor changes with a phasedelayed with respect to the external magnetic field due to the influenceof the induction magnetic field, and it is shown that the calculationresult closely matches the tendency of the exact solution.

FIG. 12 is a graph showing a phase error and an amplitude error ofcalculation values, and shows a phase error ε_(phase) and an amplitudeerror ε_(amplitude) of a magnetic field at the gravity center positionof a particle on the x axis. The phase error ε_(phase) and the amplitudeerror ε_(amplitude) are defined as follows.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 61} \right\rbrack & \; \\{{ɛ_{phase} = \frac{\theta_{exact} - \theta_{MBM}}{2\; \pi}},{ɛ_{amplitude} = \frac{H_{exact} - H_{MBM}}{H_{exact}}}} & \;\end{matrix}$

Here, θ_(exact) is a phase of an exact solution, and θ_(MEM) is a phaseof an analysis result. Furthermore, H_(exact) is a magnetic fieldobtained by the exact solution, and H_(MBM) is a magnetic field of theanalysis result. The phase error is obtained by comparing the phaseθ_(exact) when H_(exact) is maximal with the phase θ_(MBM) when H_(MBM)is maximal. From FIG. 12, the phase error between the analysis result onthe x axis and the magnetic field of the exact solution is a maximum of5.86%, the amplitude error is 6.38%, and it is shown that ahigh-accuracy analysis result is obtained compared to the exactsolution.

Series Representation of Legendre Functions

Special cases of the Legendre functions and the associated Legendrefunctions are listed below (x: =cos θ_(j)).

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 62} \right\rbrack & \; \\{{P_{0}(x)} = 1} & \left( {Q \cdot 42} \right) \\{{P_{1}(x)} = x} & \left( {Q \cdot 43} \right) \\{{P_{2}(x)} = {\frac{1}{2}\left( {{3x^{2}} - 1} \right)}} & \left( {Q \cdot 44} \right) \\{{P_{3}(x)} = {\frac{1}{2}\left( {{5x^{3}} - {3x}} \right)}} & \left( {Q \cdot 45} \right) \\{{P_{4}(x)} = {\frac{1}{8}\left( {{35x^{4}} - {30x^{2}} + 3} \right)}} & \left( {Q \cdot 46} \right) \\{{{P_{5}(x)} = {\frac{1}{8}\left( {{63x^{5}} - {70x^{2}} + {15x}} \right)}}\mspace{14mu}} & \left( {Q \cdot 47} \right) \\{and} & \left( {Q \cdot 48} \right) \\{{P_{1}^{1}(x)} = {- \left( {1 - x^{2}} \right)^{1/2}}} & \left( {Q \cdot 49} \right) \\{{P_{3}^{1}(x)} = {{- \frac{3}{2}}\left( {1 - x^{2}} \right)^{1/2}\left( {{5x^{2}} - 1} \right)}} & \left( {Q \cdot 50} \right) \\{{P_{3}^{2}(x)} = {15\left( {1 - x^{2}} \right)x}} & \left( {Q \cdot 51} \right) \\{{P_{3}^{3}(x)} = {{- 15}\left( {1 - x^{2}} \right)^{3/2}}} & \left( {Q \cdot 52} \right) \\{{P_{5}^{1}(x)} = {{- \frac{1}{8}}\left( {1 - x^{2}} \right)^{1/2}\left( {{315x^{4}} - {210x^{2}} + 15} \right)}} & \left( {Q \cdot 53} \right)\end{matrix}$

Derivation of Force Applied to Particle

The Lagrangean describing interaction between a particle (electron andnucleus) and a particle magnetic field is as follows.

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 63} \right\rbrack} & \; \\{\mathcal{L}_{MD} = {{\sum\limits_{p}^{N}\; \left\lbrack {{\frac{1}{2}M_{p}V_{p}^{2}} - {\Phi \left( x_{p} \right)} - {e\; {{A\left( x_{p} \right)} \cdot V_{p}}}} \right\rbrack} + {\sum\limits_{e}^{N}\left\lbrack {{\frac{1}{2}m_{e}v_{e}^{2}} + {e\; {{A\left( x_{e} \right)} \cdot v_{e}}}} \right\rbrack}}} & \;\end{matrix}$

The relationship between the vector potential

A  [Equation 64]

and an electromagnetic field is expressed as follows.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 65} \right\rbrack & \; \\{{B = {\nabla{\times A}}},{E = {- \frac{\partial A}{\partial t}}}} & \;\end{matrix}$

If the following relationship is used,

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 66} \right\rbrack & \; \\{{\frac{A}{t} = {\frac{\partial A}{\partial t} + {\left( {v_{i} \cdot \nabla} \right)A}}},{{\nabla\left( {v_{i} \cdot A} \right)} = {{\left( {v_{i} \cdot \nabla} \right)A} + {v_{i} \times {\nabla{\times A}}}}}} & \;\end{matrix}$

the following equation of motion is obtained.

[Equation  67]${m_{e}\frac{v_{e}}{t}} = {{{ev}_{e} \times B} + {e\; E\mspace{14mu} {for}\mspace{14mu} {electron}}}$${M_{p}\frac{V_{p}}{t}} = {{{- e}\; V_{p} \times B} - {eE} - {{\nabla{\Phi \left( x_{p} \right)}}\mspace{14mu} {for}\mspace{14mu} {nucleus}}}$

The equation of motion for an atom M=M_(p)+m_(e) is as follows.

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 68} \right\rbrack} & \; \\{{{M\frac{V}{t}} = {{{e\left( {v_{e} - V_{p}} \right)} \times {B\left( x_{i} \right)}} - {\nabla{\Phi \left( x_{i} \right)}} - {{e\left( {\left( {x_{e} - x_{p}} \right) \cdot \nabla} \right)}{E\left( x_{i} \right)}}}},\mspace{79mu} {x_{i} = \frac{{m_{e}x_{e}} + {M_{p}x_{p}}}{M_{p} + m_{e}}}} & \left( {R{.3}} \right)\end{matrix}$

With regard to the right side of the equation, by replacing the Lorentzforce with an average current density

j  [Equation 69]

and replacing the electric moment with

p _(c)  [Equation 70]

, the following expression is defined.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 71} \right\rbrack & \; \\{{M\frac{V_{i}}{t}} = {{j \times {B\left( x_{i} \right)}\delta \; V_{i}} - {\left( {p_{e} \cdot \nabla} \right)E} - {\nabla{\Phi \left( x_{i} \right)}}}} & \left( {R{.4}} \right)\end{matrix}$

From the Maxwell's equation and Q-24,

[Equation  72] ${j = {\nabla{\times H}}},{H = \left\{ \begin{matrix}{H_{o} - {\frac{1}{3}{M(H)}}} & {{NONLINER}\mspace{14mu} {MATERIAL}} \\{\frac{3\mu_{o}}{\mu + {2\mu_{o}}}H_{o}} & {{LINEAR}\mspace{14mu} {MATERIAL}}\end{matrix} \right.}$

the force applied to the atom in the end is as follows if electricpolarization is neglected.

f(x _(i))=(∇×H)×B(x _(i))δV _(i)−∇Φ(x _(i))  [Equation 73]

The force applied to the particle is obtained directly from theMaxwell's stress tensor.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 74} \right\rbrack & \; \\{f_{\alpha} = {{\frac{\partial}{\partial x_{\beta}}T^{\alpha\beta}} = \left\lbrack {\nabla{\times H \times B}} \right\rbrack_{\alpha}}} & \left( {R{.6}} \right) \\{T_{\alpha\beta} = {{\frac{1}{\mu_{o}}B_{\alpha}B_{\beta}} - {\frac{1}{2\mu_{o}}{B \cdot B}\; \delta_{\alpha\beta}}}} & \left( {R{.7}} \right)\end{matrix}$

The moment created by a localized current at the sphere is expressed asfollows.

m  [Equation 75]

The force applied to the moment is as follows to the smallest order.

f=(m×∇)×B=∇(m·B)−m(∇·B)=∇(m·B)  [Equation 76]

Therefore, the potential energy is expressed as follows together withthe potential energy due to electric polarization.

U=−m·B+p _(e) ·E  [Equation 77]

This does not include an eddy current or a true current.

[Equation  78] $\begin{matrix}{\begin{matrix}{j = {\nabla{\times \left( {H_{o} - {\frac{1}{3}M}} \right)}}} & {\left( {R{.8}} \right)} \\{= {{\nabla{\times H_{o}}} - {\frac{1}{3}\frac{{M(H)}}{H}{\nabla{\times H}}}}} & {\left( {R{.9}} \right)}\end{matrix}\begin{matrix}{{{\therefore j} = {\frac{1}{1 + {/3}}{\nabla{\times H_{o}}}}}} & {\; \left( {R{.10}} \right)}\end{matrix}\mspace{346mu} \left( {R{.11}} \right)} & \square\end{matrix}$

As a result, by calculating

∇×H _(o)  [Equation 79]

, the magnetic force applied to the magnetic sphere is obtained.

Current Due to Magnetization

Although it is difficult to completely separate

j _(m)  [Equation 80]

and the eddy current

j _(eddy)  [Equation 81]

, the following expressions are respectively defined.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 82} \right\rbrack & \; \\{{j_{m}\left( x_{p} \right)} = {\frac{1}{1 + {/3}}{\nabla{\times H_{m}}}}} & \left( {R{.12}} \right) \\{{j_{eddy}\left( x_{p} \right)} = {\frac{1}{1 + {/3}}{\nabla{\times H_{eddy}}}}} & \left( {R{.13}} \right)\end{matrix}$

Calculation of Differential Coefficients

The following is the required differential coefficients. Based on thefollowing expressions,

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 83} \right\rbrack} & \; \\{\mspace{79mu} {{\frac{}{{\cos}\; \theta}{P_{{2n} + 1}\left( {\cos \; \theta} \right)}} = {{- \frac{1}{\sin \; \theta}}{P_{{2n} + 1}^{1}\left( {\cos \; \theta} \right)}}}} & \left( {R{.14}} \right) \\{\mspace{79mu} {{\frac{}{{\cos}\; \theta}{P_{{2n} + 1}^{1}\left( {\cos \; \theta} \right)}} = {{\cot \; \theta \; {P_{{2n} + 1}^{1}\left( {\cos \; \theta} \right)}} - {\frac{1}{\sin \; \theta}{P_{{2n} + 1}^{2}\left( {\cos \; \theta} \right)}}}}} & \left( {R{.15}} \right) \\{\mspace{79mu} {{\frac{\partial}{\partial r}\left( \frac{r_{<}^{{2n} + 1}}{r_{>}^{{2n} + 2}} \right)} = \left\{ \begin{matrix}{{- \left( {{2n} + 2} \right)}\frac{a^{{2n} + 1}}{r^{{2n} + 3}}} & {r > a} \\{\left( {{2n} + 1} \right)\frac{r^{2n}}{a^{{2n} + 2}}} & {r < a}\end{matrix} \right.}} & \left( {R{.16}} \right) \\{{\frac{\partial}{\partial r}\begin{Bmatrix}{\frac{1}{r^{3}}\left( \frac{a}{r} \right)^{2n}} & {r > a} \\{{- \left( \frac{{2n} + 2}{{2n} + 1} \right)}\frac{1}{a^{3}}\left( \frac{r}{a} \right)^{2n}} & {r < a}\end{Bmatrix}} = \begin{Bmatrix}{{- \left( {{2n} + 3} \right)}\frac{1}{r^{4}}\left( \frac{a}{r} \right)^{2n}} & {r > a} \\{{- 2}{n\left( \frac{{2n} + 2}{{2n} + 1} \right)}\frac{1}{a^{4}}\left( \frac{r}{a} \right)^{{2n} - 1}} & {r < a}\end{Bmatrix}} & \left( {R{.17}} \right)\end{matrix}$

listing of only r_(pj)>a (here, since

P _(2n+1) ¹ ,P _(2n+1) ²  [Equation 84]

is proportional to sin θ, there is no divergence at θ=0) defines thefollowing expressions.

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 85} \right\rbrack} & \; \\{\frac{\partial{H_{r}\left( r_{pj} \right)}}{\partial r_{pj}} = {{- \frac{m_{jz}}{\pi}}{\sum\limits_{n = 0}^{\infty}\; {\frac{\left( {- 1} \right)^{n}{\left( {{2n} + 1} \right)!!}\left( {n + 1} \right)}{2^{n}{n!}}\frac{1}{r_{pj}^{4}}\left( \frac{a}{r_{pj}} \right)^{2n}{P_{{2n} + 1}\left( {\cos \; \theta_{pj}} \right)}}}}} & \left( {R{.18}} \right) \\{\frac{\partial{H_{r}\left( r_{pj} \right)}}{r_{pj}{\partial\cos}\; \theta_{pj}} = {{- \frac{m_{jz}}{2\pi}}{\sum\limits_{n = 0}^{\infty}\; {\frac{\left( {- 1} \right)^{n}{\left( {{2n} + 1} \right)!!}}{2^{n}{n!}}\frac{1}{r_{pj}^{4}}\left( \frac{a}{r_{pj}} \right)^{2n}\frac{P_{{2n} + 1}^{1}\left( {\cos \; \theta_{pj}} \right)}{\sin \; \theta_{pj}}}}}} & \left( {R{.19}} \right) \\{\frac{\partial{H_{\theta}\left( r_{pj} \right)}}{\partial r_{pj}} = {\frac{m_{jz}}{4\pi}{\sum\limits_{n = 0}^{\infty}\; {\frac{\left( {- 1} \right)^{n}{\left( {{2n} + 1} \right)!!}\left( {{2n} + 3} \right)}{2^{n}{\left( {n + 1} \right)!}}\frac{1}{r_{pj}^{4}}\left( \frac{a}{r_{pj}} \right)^{2n}{P_{{2n} + 1}^{1}\left( {\cos \; \theta_{pj}} \right)}}}}} & \left( {R{.20}} \right) \\{\frac{\partial{H_{\theta}\left( r_{pj} \right)}}{r_{pj}{\partial\cos}\; \theta_{pj}} = {{- \frac{m_{jz}}{4\pi}}{\underset{n = 0}{\overset{\infty}{\sum}}{\frac{\left( {- 1} \right)^{n}{\left( {{2n} + 1} \right)!!}}{2^{n}{\left( {n + 1} \right)!}}\frac{1}{r_{pj}^{4}}{\left( \frac{a}{r_{pj}} \right)^{2n}\left\lbrack {{\cot \; \theta_{pj}P_{{2n} + 1}^{1}} - \frac{P_{{2n} + 1}^{2}\left( {\cos \; \theta_{pj}} \right)}{\sin \; \theta_{pj}}} \right\rbrack}}}}} & \left( {R{.21}} \right) \\{\mspace{79mu} {\frac{\partial{H_{r}\left( {r_{pj},\theta_{pj}} \right)}}{\partial x_{p}} = {{\frac{\partial H_{r}}{\partial r_{pj}}\frac{x_{pj}}{r_{pj}}} - {\frac{\partial H_{r}}{r_{pj}{\partial\cos}\; \theta_{pj}}\frac{x_{pj}z_{pj}}{r_{pj}^{2}}}}}} & \left( {R{.22}} \right) \\{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 86} \right\rbrack} & \; \\{\mspace{79mu} {\frac{\partial{H_{r}\left( {r_{pj},\theta_{pj}} \right)}}{\partial y_{p}} = {{\frac{\partial H_{r}}{\partial r_{pj}}\frac{y_{pj}}{r_{pj}}} - {\frac{\partial H_{r}}{r_{pj}{\partial\cos}\; \theta_{pj}}\frac{y_{pj}z_{pj}}{r_{pj}^{2}}}}}} & \left( {R{.23}} \right) \\{\mspace{79mu} {\frac{\partial{H_{r}\left( {r_{pj},\theta_{pj}} \right)}}{\partial z_{p}} = {{\frac{\partial H_{r}}{\partial r_{pj}}\frac{z_{pj}}{r_{pj}}} + {\frac{\partial H_{r}}{r_{pj}{\partial\cos}\; \theta_{pj}}\left( {1 - \frac{x_{pj}z_{pj}}{r_{pj}^{2}}} \right)}}}} & \left( {R{.24}} \right) \\{\mspace{79mu} {\frac{\partial{H_{\theta}\left( {r_{pj},\theta_{pj}} \right)}}{\partial x_{p}} = {{\frac{\partial H_{\theta}}{\partial r_{pj}}\frac{x_{pj}}{r_{pj}}} - {\frac{\partial H_{\theta}}{r_{pj}{\partial\cos}\; \theta_{pj}}\frac{x_{pj}z_{pj}}{r_{pj}^{2}}}}}} & \left( {R{.25}} \right) \\{\mspace{79mu} {\frac{\partial{H_{\theta}\left( {r_{pj},\theta_{pj}} \right)}}{\partial y_{p}} = {{\frac{\partial H_{\theta}}{\partial r_{pj}}\frac{y_{pj}}{r_{pj}}} - {\frac{\partial H_{\theta}}{r_{pj}{\partial\cos}\; \theta_{pj}}\frac{y_{pj}z_{pj}}{r_{pj}^{2}}}}}} & \left( {R{.26}} \right) \\{\mspace{79mu} {\frac{\partial{H_{\theta}\left( {r_{pj},\theta_{pj}} \right)}}{\partial z_{p}} = {{\frac{\partial H_{\theta}}{\partial r_{pj}}\frac{z_{pj}}{r_{pj}}} + {\frac{\partial H_{\theta}}{r_{pj}{\partial\cos}\; \theta_{pj}}\left( {1 - \frac{x_{pj}z_{pj}}{r_{pj}^{2}}} \right)}}}} & \left( {R{.27}} \right)\end{matrix}$

Calculation of Magnetization Current Created by Magnetic Moment when theMagnetization is Parallel to the Z Axis

The following is the magnetic field when the magnetization is parallelto the z axis.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 87} \right\rbrack & \; \\{{\rho_{pj} = \sqrt{x_{pj}^{2} + y_{pj}^{2}}},{{\cos \; \theta_{pj}} = {z_{pj}/r_{pj}}}} & \left( {R{.28}} \right) \\{{H_{x}\left( {r_{pj};m_{jz}} \right)} = {\left\{ {{{H_{r}\left( {r_{pj};m_{z}} \right)}\frac{\rho_{pj}}{r_{pj}}} + {{H_{\theta}\left( {r_{pj};m_{z}} \right)}\frac{z_{pj}}{r_{pj}}}} \right\} \frac{x_{pj}}{\rho_{pj}}}} & \left( {R{.29}} \right) \\{{H_{y}\left( {r_{pj};m_{jz}} \right)} = {\left\{ {{{H_{r}\left( {r_{pj};m_{z}} \right)}\frac{\rho_{pj}}{r_{pj}}} + {{H_{\theta}\left( {r_{pj};m_{z}} \right)}\frac{z_{pj}}{r_{pj}}}} \right\} \frac{y_{pj}}{\rho_{pj}}}} & \left( {R{.30}} \right) \\{{H_{z}\left( {r_{pj};m_{jz}} \right)} = {{{H_{r}\left( {r_{pj};m_{z}} \right)}\frac{z_{pj}}{r_{pj}}} - {{H_{\theta}\left( {r_{pj};m_{z}} \right)}\frac{\rho_{pj}}{r_{pj}}}}} & \left( {R{.31}} \right)\end{matrix}$

The magnetization current is expressed as follows.

j _(m)  [Equation 88]

Then, the magnetic force created by the magnetization

f(r _(pj))m  [Equation 89]

is

f _(m)(r _(pj))=j _(m) ×B  [Equation 90]

At first,

j _(m)(r _(pj))  [Equation 91]

is obtained.

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 92} \right\rbrack} & \; \\{\mspace{79mu} {{j^{x}\left( {r_{pj},m_{jz}} \right)} = {\frac{1}{1 + {/3}}\left\{ {\frac{\partial H_{m}^{z}}{\partial y_{p}} - \frac{\partial H_{m}^{y}}{\partial z_{p}}} \right\}}}} & \left( {R{.32}} \right) \\{\frac{\partial H_{z}}{\partial y_{p}} = {\frac{1}{r_{pj}}\begin{bmatrix}{{\frac{\partial{H_{r}\left( {r_{pj};m_{jz}} \right)}}{\partial y_{p}}z_{pj}} - {{H_{r}\left( {r_{pj};m_{jz}} \right)}\frac{y_{pj}z_{pj}}{r_{pj}^{2}}} -} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jz}} \right)}}{\partial y_{p}}\rho_{pj}} - {{H_{\theta}\left( {r_{pj};m_{jz}} \right)}{y_{pj}\left( {\frac{1}{\rho_{pj}} - \frac{\rho_{pj}}{r_{pj}^{2}}} \right)}}}\end{bmatrix}}} & \left( {R{.32}\text{-}1} \right) \\{\frac{\partial H_{y}}{\partial z_{p}} = {\frac{y_{pj}}{r_{pj}}\begin{bmatrix}{\frac{\partial{H_{r}\left( {r_{pj};m_{jz}} \right)}}{\partial z_{p}} - {{H_{r}\left( {r_{pj};m_{jz}} \right)}\frac{z_{pj}}{r_{pj}^{2}}} +} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jz}} \right)}}{\partial z_{p}}\frac{z_{pj}}{\rho_{pj}}} + {{H_{\theta}\left( {r_{pj};m_{jz}} \right)}\frac{1}{\rho_{pj}}\left( {1 - \frac{z_{pj}^{2}}{r_{pj}}} \right)}}\end{bmatrix}}} & \left( {R{.32}\text{-}2} \right) \\{\mspace{79mu} {{j^{y}\left( {r_{pj},m_{jz}} \right)} = {\frac{1}{1 + {/3}}\left\{ {\frac{\partial H_{m}^{x}}{\partial z_{p}} - \frac{\partial H_{m}^{z}}{\partial x_{p}}} \right\}}}} & \left( {R{.33}} \right) \\{\frac{\partial H_{x}}{\partial z_{p}} = {\frac{x}{r_{pj}}\begin{bmatrix}{\frac{\partial{H_{r}\left( {r_{pj};m_{jz}} \right)}}{\partial z_{p}} - {{H_{r}\left( {r_{pj};m_{jz}} \right)}\frac{z_{pj}}{r_{pj}^{2}}} +} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jz}} \right)}}{\partial z_{p}}\frac{z_{pj}}{\rho_{pj}}} + {{H_{\theta}\left( {r_{pj};m_{jz}} \right)}\frac{1}{\rho_{pj}}\left( {1 - \frac{z_{pj}^{2}}{r_{pj}^{2}}} \right)}}\end{bmatrix}}} & \left( {R{.33}\text{-}1} \right) \\{\frac{\partial H_{z}}{\partial x_{p}} = {\frac{1}{r_{pj}}\begin{bmatrix}{{\frac{\partial{H_{r}\left( {r_{pj};m_{jz}} \right)}}{\partial x_{p}}z_{pj}} - {{H_{r}\left( {r_{pj};m_{jz}} \right)}\frac{x_{pj}z_{pj}}{r_{pj}^{2}}} -} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jz}} \right)}}{\partial x_{p}}\rho_{pj}} - {{H_{\theta}\left( {r_{pj};m_{jz}} \right)}{x_{pj}\left( {\frac{1}{\rho_{pj}} - \frac{\rho_{pj}}{r_{pj}^{2}}} \right)}}}\end{bmatrix}}} & \left( {R{.33}\text{-}2} \right) \\{\mspace{79mu} {{j^{z}\left( {r_{pj},m_{jz}} \right)} = {\frac{1}{1 + {/3}}\left\{ {\frac{\partial H_{m}^{y}}{\partial x_{p}} - \frac{\partial H_{m}^{x}}{\partial y_{p}}} \right\}}}} & \left( {R{.34}} \right) \\{\frac{\partial H_{x}}{\partial y_{p}} = {\frac{x_{pj}}{r_{pj}}\begin{bmatrix}{\frac{\partial{H_{r}\left( {r_{pj};m_{jz}} \right)}}{\partial y_{p}} - {{H_{r}\left( {r_{pj};m_{jz}} \right)}\frac{y_{pj}}{r_{pj}^{2}}} +} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jz}} \right)}}{\partial y_{p}}\frac{z_{pj}}{\rho_{pj}}} - {{H_{\theta}\left( {r_{pj};m_{jz}} \right)}\frac{y_{pj}z_{pj}}{\rho_{pj}}\left( {\frac{1}{r_{pj}^{2}} + \frac{1}{\rho_{pj}^{2}}} \right)}}\end{bmatrix}}} & \left( {R{.34}\text{-}1} \right) \\{\frac{\partial H_{y}}{\partial x_{p}} = {+ {\frac{y_{pj}}{r_{pj}}\begin{bmatrix}{\frac{\partial{H_{r}\left( {r_{pj};m_{jz}} \right)}}{\partial x_{p}} - {{H_{r}\left( {r_{pj};m_{jz}} \right)}\frac{x_{pj}}{r_{pj}^{2}}} +} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jz}} \right)}}{\partial x_{p}}\frac{z_{pj}}{\rho_{pj}}} - {{H_{\theta}\left( {r_{pj};m_{jz}} \right)}\frac{x_{pj}z_{pj}}{\rho_{pj}}\left( {\frac{1}{r_{pj}^{2}} + \frac{1}{\rho_{pj}^{2}}} \right)}}\end{bmatrix}}}} & \left( {R{.34}\text{-}2} \right)\end{matrix}$

When the Magnetization is Parallel to the x Axis

The following is the magnetic field when the magnetization is parallelto the x axis.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 93} \right\rbrack & \; \\{{\rho_{pj} = \sqrt{y_{pj}^{2} + z_{pj}^{2}}},{{\cos \; \theta_{pj}} = {x_{pj}/r_{pj}}}} & \left( {R{.34}} \right) \\\left\lbrack {{Equation}\mspace{14mu} 94} \right\rbrack & \; \\{{H_{x}\left( {r_{pj};m_{jx}} \right)} = {{{H_{r}\left( {r_{pj};m_{x}} \right)}\frac{x_{pj}}{r_{pj}}} - {{H_{\theta}\left( {r_{pj};m_{x}} \right)}\frac{\rho_{pj}}{r_{pj}}}}} & \left( {R{.35}} \right) \\{{H_{y}\left( {r_{pj};m_{jx}} \right)} = {\left\{ {{{H_{r}\left( {r_{pj};m_{x}} \right)}\frac{\rho_{pj}}{r_{pj}}} + {{H_{\theta}\left( {r_{pj};m_{x}} \right)}\frac{x_{pj}}{r_{pj}}}} \right\} \frac{y_{pj}}{\rho_{pj}}}} & \left( {R{.36}} \right) \\{{H_{z}\left( {r_{pj};m_{jx}} \right)} = {\left\{ {{{H_{r}\left( {r_{pj};m_{x}} \right)}\frac{\rho_{pj}}{r_{pj}}} + {{H_{\theta}\left( {r_{pj};m_{x}} \right)}\frac{x_{pj}}{r_{pj}}}} \right\} \frac{z_{pj}}{\rho_{pj}}}} & \left( {R{.37}} \right)\end{matrix}$

The magnetization current

j _(m) _(x)   [Equation 95]

is obtained.

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 96} \right\rbrack} & \; \\{\mspace{79mu} {{j^{x}\left( {r_{pj},m_{jx}} \right)} = {\frac{1}{1 + {/3}}\left\{ {\frac{\partial H_{m}^{z}}{\partial y_{p}} - \frac{\partial H_{m}^{y}}{\partial z_{p}}} \right\}}}} & \left( {R{.38}} \right) \\{\frac{\partial H_{z}}{\partial y_{p}} = {\frac{z_{pj}}{r_{pj}}\begin{bmatrix}{\frac{\partial{H_{r}\left( {r_{pj};m_{jx}} \right)}}{\partial y_{p}} - {{H_{r}\left( {r_{pj};m_{jx}} \right)}\frac{y_{pj}}{r_{pj}^{2}}} +} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jx}} \right)}}{\partial y_{p}}\frac{x_{pj}}{\rho_{pj}}} - {{H_{\theta}\left( {r_{pj};m_{jx}} \right)}\frac{x_{pj}y_{pj}}{\rho_{pj}}\left( {\frac{1}{r_{pj}^{2}} + \frac{1}{\rho_{pj}^{2}}} \right)}}\end{bmatrix}}} & \left( {R{.38}\text{-}1} \right) \\{\frac{\partial H_{y}}{\partial z_{p}} = {\frac{y_{pj}}{r_{pj}}\begin{bmatrix}{\frac{\partial{H_{r}\left( {r_{pj};m_{jx}} \right)}}{\partial z_{p}} - {{H_{r}\left( {r_{pj};m_{jx}} \right)}\frac{z_{pj}}{r_{pj}^{2}}} +} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jx}} \right)}}{\partial z_{p}}\frac{x_{pj}}{\rho_{pj}}} - {{H_{\theta}\left( {r_{pj};m_{jx}} \right)}\frac{x_{pj}z_{pj}}{\rho_{pj}^{x}}{z_{pj}\left( {\frac{1}{r_{pj}^{2}} + \frac{1}{\rho_{pj}^{2}}} \right)}}}\end{bmatrix}}} & \left( {R{.38}\text{-}2} \right) \\{\mspace{79mu} {{j^{y}\left( {r_{pj},m_{jx}} \right)} = {\frac{1}{1 + {/3}}\left\{ {\frac{\partial H_{m}^{x}}{\partial z_{p}} - \frac{\partial H_{m}^{z}}{\partial x_{p}}} \right\}}}} & \left( {R{.39}} \right) \\{\frac{\partial H_{x}}{\partial z_{p}} = {\frac{1}{r_{pj}}\begin{bmatrix}{{\frac{\partial{H_{r}\left( {r_{pj};m_{jx}} \right)}}{\partial z_{p}}x_{pj}} - {{H_{r}\left( {r_{pj};m_{jx}} \right)}\frac{x_{pj}z_{pj}}{r_{pj}^{2}}} -} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jx}} \right)}}{\partial z_{p}}\rho_{pj}} - {{H_{\theta}\left( {r_{pj};m_{jx}} \right)}{z_{pj}\left( {\frac{1}{\rho_{pj}} - \frac{\rho_{pj}}{r_{pj}^{2}}} \right)}}}\end{bmatrix}}} & \left( {R{.39}\text{-}1} \right) \\{\frac{\partial H_{z}}{\partial x_{p}} = {\frac{1}{r_{pj}}\begin{bmatrix}{{\frac{\partial{H_{r}\left( {r_{pj};m_{jx}} \right)}}{\partial x_{p}}z_{pj}} + {{H_{r}\left( {r_{pj};m_{jx}} \right)}\left( {1 - \frac{x_{pj}z_{pj}}{r_{pj}^{2}}} \right)} +} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jx}} \right)}}{\partial x_{p}}\frac{x_{pj}z_{pj}}{\rho_{pj}}} + {{H_{\theta}\left( {r_{pj};m_{jx}} \right)}\frac{z_{pj}}{\rho_{pj}}\left( {1 - \frac{x_{pj}^{2}}{r_{pj}^{2}}} \right)}}\end{bmatrix}}} & \left( {R{.39}\text{-}2} \right) \\{\mspace{79mu} {{j^{z}\left( {r_{pj},m_{jx}} \right)} = {\frac{1}{1 + {/3}}\left\{ {\frac{\partial H_{m}^{y}}{\partial x_{p}} - \frac{\partial H_{m}^{x}}{\partial y_{p}}} \right\}}}} & \left( {R{.40}} \right) \\{\frac{\partial H_{x}}{\partial y_{p}} = {\frac{1}{r_{pj}}\begin{bmatrix}{{\frac{\partial{H_{r}\left( {r_{pj};m_{jx}} \right)}}{\partial y_{p}}x_{pj}} - {{H_{r}\left( {r_{pj};m_{jx}} \right)}\frac{x_{pj}y_{pj}}{r_{pj}^{2}}} -} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jx}} \right)}}{\partial y_{p}}\rho_{pj}} - {{H_{\theta}\left( {r_{pj};m_{jx}} \right)}{y_{pj}\left( {\frac{1}{\rho_{pj}} - \frac{\rho_{pj}}{r_{pj}^{2}}} \right)}}}\end{bmatrix}}} & \left( {R{.40}\text{-}1} \right) \\{\frac{\partial H_{y}}{\partial x_{p}} = {\frac{y_{pj}}{r_{pj}}\begin{bmatrix}{\frac{\partial{H_{r}\left( {r_{pj};m_{jx}} \right)}}{\partial x_{p}} - {{H_{r}\left( {r_{pj};m_{jx}} \right)}\frac{x_{pj}}{r_{pj}^{2}}} +} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jx}} \right)}}{\partial x_{p}}\frac{x_{pj}}{\rho_{pj}}} + {{H_{\theta}\left( {r_{pj};m_{jx}} \right)}\frac{1}{\rho_{pj}}\left( {1 - \frac{x_{pj}^{2}}{r_{pj}^{2}}} \right)}}\end{bmatrix}}} & \left( {R{.40}\text{-}2} \right)\end{matrix}$

When the Magnetization is Parallel to the y Axis

The following is the magnetic field when the magnetization is parallelto the y axis.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 97} \right\rbrack & \; \\{{\rho_{pj} = \sqrt{x_{pj}^{2} + z_{pj}^{2}}},{{\cos \; \theta_{pj}} = {y_{pj}/r_{pj}}}} & \left( {R{.41}} \right) \\{{H_{x}\left( {r_{pj};m_{jy}} \right)} = {\left\{ {{{H_{r}\left( {r_{pj};m_{y}} \right)}\frac{\rho_{pj}}{r_{pj}}} + {{H_{\theta}\left( {r_{pj};m_{y}} \right)}\frac{y_{pj}}{r_{pj}}}} \right\} \frac{x_{pj}}{\rho_{pj}}}} & \left( {R{.42}} \right) \\{{H_{y}\left( {r_{pj};m_{jy}} \right)} = {{{H_{r}\left( {r_{pj};m_{y}} \right)}\frac{y_{pj}}{r_{pj}}} - {{H_{\theta}\left( {r_{pj};m_{y}} \right)}\frac{\rho_{pj}}{r_{pj}}}}} & \left( {R{.43}} \right) \\{{H_{z}\left( {r_{pj};m_{jy}} \right)} = {\left\{ {{{H_{r}\left( {r_{pj};m_{y}} \right)}\frac{\rho_{pj}}{r_{pj}}} + {{H_{\theta}\left( {r_{pj};m_{y}} \right)}\frac{y_{pj}}{r_{pj}}}} \right\} \frac{z_{pj}}{\rho_{pj}}}} & \left( {R{.44}} \right)\end{matrix}$

The magnetization current

j _(m) _(y)   [Equation 98]

is obtained.

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 99} \right\rbrack} & \; \\{\mspace{79mu} {{j^{x}\left( {r_{pj},m_{jy}} \right)} = {\frac{1}{1 + {/3}}\left\{ {\frac{\partial H_{m}^{z}}{\partial y_{p}} - \frac{\partial H_{m}^{y}}{\partial z_{p}}} \right\}}}} & \left( {R{.45}} \right) \\{\frac{\partial H_{z}}{\partial y_{p}} = {\frac{z_{pj}}{r_{pj}}\begin{bmatrix}{\frac{\partial{H_{r}\left( {r_{pj};m_{jy}} \right)}}{\partial y_{p}} - {{H_{r}\left( {r_{pj};m_{jy}} \right)}\frac{y_{pj}}{r_{pj}^{2}}} +} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jy}} \right)}}{\partial y_{p}}\frac{y_{pj}}{\rho_{pj}}} + {{H_{\theta}\left( {r_{pj};m_{jy}} \right)}\frac{1}{\rho_{pj}}\left( {1 - \frac{y_{pj}^{2}}{r_{pj}^{2}}} \right)}}\end{bmatrix}}} & \left( {R{.45}\text{-}1} \right) \\{\frac{\partial H_{y}}{\partial z_{p}} = {\frac{1}{r_{pj}}\begin{bmatrix}{{\frac{\partial{H_{r}\left( {r_{pj};m_{jy}} \right)}}{\partial z_{p}}y_{pj}} - {{H_{r}\left( {r_{pj};m_{jy}} \right)}\frac{y_{pj}z_{pj}}{r_{pj}^{2}}} -} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jy}} \right)}}{\partial z_{p}}\rho_{pj}} - {{H_{\theta}\left( {r_{pj};m_{jy}} \right)}{z_{pj}\left( {\frac{1}{\rho_{pj}} - \frac{\rho_{pj}}{r_{pj}^{2}}} \right)}}}\end{bmatrix}}} & \left( {R{.45}\text{-}2} \right) \\{\mspace{79mu} {{j^{y}\left( {r_{pj},m_{jy}} \right)} = {\frac{1}{1 + {/3}}\left\{ {\frac{\partial H_{m}^{x}}{\partial z_{p}} - \frac{\partial H_{m}^{z}}{\partial x_{p}}} \right\}}}} & \left( {R{.46}} \right) \\{\frac{\partial H_{x}}{\partial z_{p}} = {\frac{x_{pj}}{r_{pj}}\begin{bmatrix}{\frac{\partial{H_{r}\left( {r_{pj};m_{jy}} \right)}}{\partial z_{p}} - {{H_{r}\left( {r_{pj};m_{jy}} \right)}\frac{z_{pj}}{r_{pj}^{2}}} +} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jy}} \right)}}{\partial z_{p}}\frac{y_{pj}}{\rho_{pj}}} - {{H_{\theta}\left( {r_{pj};m_{jy}} \right)}\frac{y_{pj}z_{pj}}{\rho_{pj}}\left( {\frac{1}{r_{pj}^{2}} + \frac{1}{\rho_{pj}^{2}}} \right)}}\end{bmatrix}}} & \left( {R{.46}\text{-}1} \right) \\{\frac{\partial H_{z}}{\partial x_{p}} = {\frac{z_{pj}}{r_{pj}}\begin{bmatrix}{\frac{\partial{H_{r}\left( {r_{pj};m_{jy}} \right)}}{\partial x_{p}} - {{H_{r}\left( {r_{pj};m_{jy}} \right)}\frac{x_{pj}}{r_{pj}^{2}}} +} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jy}} \right)}}{\partial x_{p}}\frac{y_{pj}}{\rho_{pj}}} - {{H_{\theta}\left( {r_{pj};m_{jy}} \right)}\frac{x_{pj}z_{pj}}{\rho_{pj}}\left( {\frac{1}{r_{pj}^{2}} + \frac{1}{\rho_{pj}^{2}}} \right)}}\end{bmatrix}}} & \left( {R{.46}\text{-}2} \right) \\{\mspace{79mu} {{j^{z}\left( {r_{pj},m_{jy}} \right)} = {\frac{1}{1 + {/3}}\left\{ {\frac{\partial H_{m}^{y}}{\partial x_{p}} - \frac{\partial H_{m}^{x}}{\partial y_{p}}} \right\}}}} & \left( {R{.47}} \right) \\{\frac{\partial H_{x}}{\partial y_{p}} = {\frac{x_{pj}}{r_{pj}}\begin{bmatrix}{\frac{\partial{H_{r}\left( {r_{pj};m_{jy}} \right)}}{\partial y_{p}} - {{H_{r}\left( {r_{pj};m_{jy}} \right)}\frac{y_{pj}}{r_{pj}^{2}}} +} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jy}} \right)}}{\partial y_{p}}\frac{y_{pj}}{\rho_{pj}}} + {{H_{\theta}\left( {r_{pj};m_{jy}} \right)}\frac{1}{\rho_{pj}}\left( {1 - \frac{y_{pj}^{2}}{r_{pj}^{2}}} \right)}}\end{bmatrix}}} & \left( {R{.47}\text{-}1} \right) \\{\frac{\partial H_{y}}{\partial x_{p}} = {\frac{1}{r_{pj}}\begin{bmatrix}{{\frac{\partial{H_{r}\left( {r_{pj};m_{jy}} \right)}}{\partial x_{p}}y_{pj}} - {{H_{r}\left( {r_{pj};m_{jy}} \right)}\frac{x_{pj}y_{pj}}{r_{pj}^{2}}} -} \\{{\frac{\partial{H_{\theta}\left( {r_{pj};m_{jy}} \right)}}{\partial x_{p}}\rho_{pj}} - {{H_{\theta}\left( {r_{pj};m_{jy}} \right)}{x_{pj}\left( {\frac{1}{\rho_{pj}} - \frac{\rho_{pj}}{r_{pj}^{2}}} \right)}}}\end{bmatrix}}} & \left( {R{.47}\text{-}2} \right)\end{matrix}$

Calculation of Eddy Current Created by Induction Magnetic Field

According to Q-21, the induction magnetic field inside the conductor isdefined as follows.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 100} \right\rbrack & \; \\{{H_{ind}\left( x_{p} \right)} = {{M_{ant}\left( x_{p} \right)} + {\frac{1}{4\pi}{\sum\limits_{j}\; \frac{{3{n\left( {n \cdot m_{ant}^{j}} \right)}} - m_{ant}^{j}}{x_{pj}^{3}}}}}} & \left( {R{.48}} \right) \\{{G\left( x_{pj} \right)} = \left\{ \begin{matrix}{\frac{1}{3}\frac{a_{j}^{3}}{x_{pj}}} & {x_{pj} > a_{j}} \\{\frac{a_{j}^{2}}{2} - \frac{x_{pj}^{2}}{6}} & {x_{pj} < a_{j}}\end{matrix} \right.} & \left( {R{.49}} \right) \\{{M_{ant}\left( x_{p} \right)} = {{- C_{1}}{\sum\limits_{j}\; {\sigma_{j}{G\left( x_{pj} \right)}\frac{\; B_{j}}{\; t}}}}} & \left( {R{.49}\text{-}1} \right) \\{m_{ant}^{j} = {C_{2}{M_{ant}\left( x_{j} \right)}\delta \; V_{j}}} & \left( {R{.49}\text{-}2} \right)\end{matrix}$

The eddy current is as follows.

     [Equation  101] $\begin{matrix}{{j_{eddy}\left( x_{p} \right)} = {\frac{1}{1 + {/3}}{\nabla{\times {H_{ind}\left( x_{p} \right)}}}}} & {\left( {R{.50}} \right)} \\{= {\frac{1}{1 + {/3}}{\nabla{\times \left\lbrack {{M_{ant}\left( x_{p} \right)} + {\frac{1}{4\pi}{\sum\limits_{j}\; \frac{{3{n\left( {n \cdot m_{ant}^{j}} \right)}} - m_{ant}^{j}}{x_{pj}^{3}}}}} \right\rbrack}}}} & {\left( {R{.51}} \right)}\end{matrix}$ $\begin{matrix}{\mspace{79mu} {{\nabla{\times {M_{ant}\left( x_{p} \right)}}} = {{- C_{1}}{\sum\limits_{j}\; {\sigma_{j}\left\lbrack {\nabla{\times \left\{ {{G\left( x_{pj} \right)}\frac{\; B_{j}}{\; t}} \right\}}} \right\rbrack}}}}} & { \left( {R{.52}\text{-}1} \right)}\end{matrix}$ $\begin{matrix}{\mspace{79mu} {\nabla{\times \left\lbrack {\frac{1}{4\pi}{\sum\limits_{j}\; \frac{{3{n\left( {n \cdot m_{ant}^{j}} \right)}} - m_{ant}^{j}}{x_{pj}^{3}}}} \right\rbrack}}} & {\mspace{245mu} \left( {R{.52}\text{-}2} \right)}\end{matrix}$

Expression R-52-2 is the same as a case where the expansion order ofspherical harmonics n=0 in R-12.

Calculation of Force Applied to Particle

By adding m_(x), m_(y), m_(z), and the contributions from the eddycurrent, the total current is obtained.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 102} \right\rbrack & \; \\{{j\left( x_{p} \right)} = {{\sum\limits_{j \neq p}\; \left\{ {{j\left( {r_{pj},m_{x}} \right)} + {j\left( {r_{pj},m_{y}} \right)} + {j\left( {r_{pj},m_{z}} \right)}} \right\}} + {j_{eddy}\left( r_{p} \right)}}} & \left( {R{.53}} \right)\end{matrix}$

The force applied to a particle at a coordinate

X _(P)  [Equation 103]

is defined as follows.

f(x _(p))=j(x _(p))×B(x _(p))δV _(p)−∇_(p)Φ(r _(p))  (R•54)

In the end, it is necessary to add a term derived from the derivative ofthe following expression.

_(EM)  [Equation 105]

This is a virtual force created by an error of the magnetic field

H _(oi) −H _(o)(x _(i))  [Equation 106]

, and the total energy of the motion and the magnetic field isconserved.

[Equation 107]

f(x _(p))=j(x _(p))×B(x _(p))δV _(p)−∇_(p)Φ(r _(p)+μ_(o)[{(H _(op) −H_(o)(x _(p)))·∇}H _(o)(x _(p))−j(x _(p))×{H _(op) −H _(o)(x _(p))}]δV_(p)  (R•55)

Another Embodiment

In the above-described embodiment, a case has been described where thedynamic magnetic field analysis is performed using the inducedmagnetization in the first term on the right side of Q-21 and Expression(Q-21) for the induction magnetic field represented by interactionbetween magnetic moments in the second term on the right side. In thisembodiment, the induction magnetic field is formulated using the exactsolution of the vector potential derived from a differential equation ofa vector potential, instead of using Expression Q-21. The followingdescription will be provided focusing on the difference from theabove-described embodiment.

With regard to a vector potential A, the following differential equationis established.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 108} \right\rbrack & \; \\{{\mu \; \sigma \frac{\partial{A(r)}}{\partial t}} = {\nabla^{2}{A(r)}}} & \left( {S\text{-}1} \right)\end{matrix}$

The solution of the differential equation is as follows.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 109} \right\rbrack & \; \\{{A(r)} = {{- \frac{\mu}{4\pi}}{\int{{^{3}r^{\prime}}\frac{{\sigma \left( r^{\prime} \right)}{\overset{.}{A}\left( r^{\prime} \right)}}{{r - r^{\prime}}}}}}} & \left( {S\text{-}2} \right)\end{matrix}$

Here, if it is assumed that

r′=r _(j)+ρ  [Equation 110]

, and if an object is divided into particles, the vector potential ofeach particle at a position r_(i) is as follows.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 111} \right\rbrack & \; \\\begin{matrix}{{A\left( r_{i} \right)} = {{- \frac{\mu}{4\pi}}{\sum\limits_{j \Subset i}^{N_{i}}\; {\int_{r_{j} + v_{j}}\ {{^{3}r^{\prime}}\frac{{\sigma \left( r^{\prime} \right)}{\overset{.}{A}\left( r^{\prime} \right)}}{{r - r^{\prime}}}}}}}} \\{= {{- \frac{\mu}{4\pi}}{\sum\limits_{j \Subset i}^{N_{i}}\; {\int_{v_{j}}\ {{^{3}\rho}\frac{\sigma_{j}{{\overset{.}{A}}_{j}(\rho)}}{{r_{ij} - \rho}}}}}}}\end{matrix} & \left( {S\text{-}3} \right)\end{matrix}$

Here, v_(j) is a volume of each particle. With regard to the sum of j,when a vector

r _(i) −r _(j)  [Equation 112]

crosses the particle surface, the sum is not taken. A condition whichshould be satisfied by the particle size a is as follows.

√{square root over (2/(μσω))}>>a  [Equation 113]

In this case, the division of each object into particles may causelosing of the transitional symmetry of the vector potential A. Thus, theinventors have found that gauge transformation shown in S-4 isintroduced, thereby recovering the transitional symmetry of the vectorpotential A. In this specification, the gauge transformation is referredto as “beads gauge”.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 114} \right\rbrack & \; \\{{A_{j}(\rho)}->{{A_{j}(\rho)} - {\frac{1}{2}B_{j} \times g_{i}}}} & \left( {S\text{-}4} \right)\end{matrix}$

Here, a vector g_(i) is a local gravity center vector relating to aparticle group (the number of particles: N_(i)) to be calculated and isexpressed as follows.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 115} \right\rbrack & \; \\{g_{i} = {\frac{1}{N_{i}}{\sum\limits_{j \Subset i}^{N_{i}}\; r_{j}}}} & \left( {S\text{-}5} \right)\end{matrix}$

Therefore, the vector g_(i) represents the gravity center position ofthe particle group. It can be said that the beads gauge shown in S-4 isdescribed using the outer product of a magnetic flux density B_(j) andthe local gravity center vector g_(i).

With regard to the beads gauge, the relationship expression of themagnetic flux density B and the vector potential A

B(r _(j))=∇×A _(j)(ρ)  [Equation 116]

is obtained as follows.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 117} \right\rbrack & \; \\{{B\left( r_{j} \right)} = {{\left( {1 - \frac{1}{N_{i}}} \right)B_{j}} = {B_{j} + {O\left( \frac{1}{N_{i}} \right)}}}} & \left( {S\text{-}6} \right)\end{matrix}$

Therefore, it can be understood that, if the number of particles N_(i)is sufficiently large, the transitional symmetry is satisfied.

Here, from the gauge transformation shown in S-4, the time differentialof the vector potential included in the right side of S-3

{dot over (A)} _(j)(ρ)  [Equation 118]

can be expressed as follows using the time differential of the magneticflux density B.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 119} \right\rbrack & \; \\{{{\overset{.}{A}}_{j}(\rho)} = {\frac{1}{2}{\overset{.}{B}}_{j} \times \left( {d_{ji} + \rho} \right)}} & \left( {S\text{-}7} \right)\end{matrix}$

If a vector d_(ji) is

[Equation 120]

d _(ji) =r _(j) −g _(i)  (S-8)

, the time differential of the vector potential is expressed as follows.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 121} \right\rbrack & \; \\{{{\overset{.}{A}}_{j}(\rho)} = {{\frac{1}{2}{\overset{.}{B}}_{j} \times \left( {r_{j} + \rho} \right)} - {\frac{1}{2}{\overset{.}{B}}_{j} \times g_{i}}}} & \left( {S\text{-}9} \right)\end{matrix}$

Therefore, if integration is executed by substituting S-9 in S-3, theexact solution of the vector potential A can be described using the timedifferential of the magnetic flux density B.

In order to derive the exact solution of the vector potential A, first,integration when only

{dot over (B)} _(z)  [Equation 122]

which is the time differential of the z component of the magnetic fluxdensity B is applied is executed. With this, the vector potential A is

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 123} \right\rbrack} & \; \\{{A\left( r_{i} \right)} = {{{- \frac{\mu}{8\pi}}{\sum\limits_{j \Subset i}^{N_{i}}\; {{\sigma_{j}\left\lbrack {{- {{\overset{.}{B}}_{jz}\left( {y_{j} - g_{iy}} \right)}},{{\overset{.}{B}}_{jz}\left( {x_{j} - g_{ix}} \right)},0} \right\rbrack}{\int_{v_{j}}{{^{3}\rho}\frac{1}{{r_{ij} - \rho}}}}}}} - {\frac{\mu}{8\pi}{\sum\limits_{j \Subset i}^{N_{i}}{\int_{v_{j}}{{^{3}\rho}\frac{\sigma_{j}\left\lbrack {{{- {\overset{.}{B}}_{jz}}\rho_{y}},{{\overset{.}{B}}_{jz}\rho_{x}},0} \right\rbrack}{{r_{ij} - \rho}}}}}}}} & \left( {S\text{-}10} \right)\end{matrix}$

If the contributions from

{dot over (B)} _(x) ,{dot over (B)} _(y)  [Equation 124]

which are the time differentials of the x component and the y componentof the magnetic flux density B are added, the following expression isobtained.

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 125} \right\rbrack} & \; \\{{A\left( r_{i} \right)} = {{- \frac{\mu}{2}}{\sum\limits_{j \Subset i}^{N_{i}}\; {{\sigma_{j}\left( {{\overset{.}{B}}_{j} \times d_{ji}} \right)}\left\{ {\begin{matrix}{{\frac{1}{3}r_{ij}^{2}} + {\frac{1}{2}\left( {a^{2} - r_{ij}^{2}} \right)}} & {r_{ij} \leq a} \\{\frac{1}{3}\frac{a^{3}}{r_{ij}}} & {r_{ij} > a}\end{matrix} - {\frac{\mu}{6} {\sum\limits_{j \Subset i}^{N_{i}}\; {{\sigma_{j}\left( {{\overset{.}{B}}_{j} \times r_{ij}} \right)}\left\{ \begin{matrix}{{\frac{1}{5}r_{ij}^{2}} + {\frac{1}{2}\left( {a^{2} - r_{ij}^{2}} \right)}} & {r_{ij} \leq a} \\{\frac{1}{5}\frac{a^{5}}{r_{ij}^{3}}} & {r_{ij} > a}\end{matrix} \right.}}}} \right.}}}} & \left( {S\text{-}11} \right)\end{matrix}$

An induction magnetic field H which describes the inside of a conductorto be analyzed from the following relationship expression

μH=B=∇×A  [Equation 126]

is

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 127} \right\rbrack} & \; \\\begin{matrix}{{H\left( r_{i} \right)} = {{- \frac{1}{6}}{\sum\limits_{j \Subset i}^{N_{i}}\; {\sigma_{j}\left\{ \begin{matrix}{{{- \left( {r_{ij} \cdot d_{ji}} \right)}{\overset{.}{B}}_{j}} + {\left( {r_{ij} \cdot {\overset{.}{B}}_{j}} \right)d_{ji}} + {O\left( {1/N_{i}} \right)}} & {r_{ij} \leq a} \\\begin{matrix}{{{- \frac{a^{3}}{r_{ij}^{3}}}\left( {r_{ij} \cdot d_{ji}} \right){\overset{.}{B}}_{j}} + {\frac{a^{3}}{r_{ij}^{3}}\left( {r_{ij} \cdot {\overset{.}{B}}_{j}} \right)d_{ji}} +} \\{O\left( {1/N_{i}} \right)}\end{matrix} & {r_{ij} > a}\end{matrix} \right.}}}} \\{{{- \frac{1}{6}}{\sum\limits_{j \Subset i}^{N_{i}}\; {\sigma_{j}\left\{ \begin{matrix}{{\left( {a^{2} - {\frac{6}{5}r_{ij}^{2}}} \right){\overset{.}{B}}_{j}} + {\frac{3}{5}\left( {r_{ij} \cdot {\overset{.}{B}}_{j}} \right)r_{ij}}} & {r_{ij} \leq a} \\{\frac{a^{5}}{5r_{ij}^{3}}\left\{ {{3\left( {n_{ij} \cdot {\overset{.}{B}}_{j}} \right)n_{ij}} - {\overset{.}{B}}_{j}} \right\}} & {r_{ij} > a}\end{matrix} \right.}}}} \\{\cong {{{- \frac{1}{6}}\sigma_{i}a^{2}{\overset{.}{B}}_{i}} - {\frac{1}{6}{\sum\limits_{j \Subset i}^{N_{i}}\; {\sigma_{j}{\frac{a^{3}}{r_{ij}^{3}}\begin{bmatrix}{{{- \left( {r_{ij} \cdot d_{ji}} \right)}{\overset{.}{B}}_{j}} + {\left( {r_{ij} \cdot {\overset{.}{B}}_{j}} \right)d_{ji}} +} \\{\frac{a^{2}}{5}\left\{ {{3\left( {n_{ij} \cdot {\overset{.}{B}}_{j}} \right)n_{ij}} - {\overset{.}{B}}_{j}} \right\}}\end{bmatrix}}}}}}}\end{matrix} & \left( {S\text{-}12} \right)\end{matrix}$

Here, by replacing with

r _(j) =r _(i) −r _(ij)  [Equation 128]

as the expression for the induction magnetic field H,

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 129} \right\rbrack} & \; \\{{H\left( r_{i} \right)} = {{{- \frac{a^{3}}{6}}{\sum\limits_{j \Subset i}^{N_{i}}\; {\sigma_{j}\left\lbrack {\frac{{\overset{.}{B}}_{j} - {\left( {n_{ij} \cdot {\overset{.}{B}}_{j}} \right)n_{ij}}}{r_{ij}} - \frac{{\left( {r_{ij} \cdot d_{ii}} \right){\overset{.}{B}}_{j}} - {\left( {r_{ij} \cdot {\overset{.}{B}}_{j}} \right)d_{ii}}}{r_{ij}^{3}}} \right\rbrack}}} - {\frac{1}{6}\sigma_{i}a^{2}{\overset{.}{B}}_{i}} - {\frac{1}{30}{\sum\limits_{j \Subset i}^{N_{i}}\; {\sigma_{j}a^{5}\frac{{3\left( {n_{ij} \cdot {\overset{.}{B}}_{j}} \right)n_{ij}} - {\overset{.}{B}}_{j}}{r_{ij}^{3}}}}}}} & \left( {S\text{-}13} \right)\end{matrix}$

is obtained.

The magnetic field calculation unit 121 calculates the inductionmagnetic field generated by the particle system using the expression forthe induction magnetic field shown in S-13. In the calculation forobtaining the time variation value of the induction magnetic field, thesame method as in the above-described embodiment should be used. Themagnetic field calculation unit 121 may calculate a current density fromthe obtained induction magnetic field. The force calculation unit 122may calculate an electromagnetic force from the values of the obtainedmagnetic field and current.

In this embodiment, the local gravity center vector g_(i) shown in S-5is introduced for particle groups at the same potential, and separatelocal gravity center vectors are introduced for a plurality of particlegroups electrically insulated from one another. For example, when aparticle system to be analyzed has a plurality of portions electricallyinsulated from one another, local gravity center vectors havingdifferent gravity center positions are applied to particle groupsforming the respective portions. When an object to be analyzed isdeformed over time and divided into a plurality of objects, and thedivided portions are insulated from one another, local gravity centervectors having different gravity center positions are applied toparticle groups forming the respective portions.

Accordingly, when a plurality of particle groups including a firstparticle group and a second particle group electrically insulated fromeach other are arranged in a particle system to be analyzed, themagnetic field calculation unit 121 performs analysis using anexpression for an induction magnetic field corresponding to eachparticle group. For example, the induction magnetic field in the firstparticle group is calculated using an expression for a first inductionmagnetic field derived by gauge transformation using a first localgravity center vector representing the gravity center position of thefirst particle group. The induction magnetic field in a second particlegroup is calculated using an expression for a second induction magneticfield derived by gauge transformation using a second local gravitycenter vector representing the gravity center position of the secondparticle group. Specifically, in the expression for the inductionmagnetic field shown in S-13, d_(ij) included in the first term on theright side takes a different value in the calculation of each particlegroup.

Accuracy verification was performed for an analysis method usingExpression (S-13) for an induction magnetic field according to thisembodiment. As in the above-described embodiment, an analysis model of aconductive sphere shown in FIG. 10 was used, and a uniform externalmagnetic field was applied in a space to be H_(ext,x)=H_(O) sin (2πft)in the x-axis direction. Calculation conditions were H_(O)=7.948×10⁴[A/m], f=100 [Hz], the radius of the conductive sphere A=10 [mm], andelectrical conductivity σ=59×10⁶ [S/m]. The number of particles (atoms)forming the conductive sphere was 6099 and the particles were arrangedin an fcc structure.

FIGS. 13A to 13C are graphs showing calculation values of a magneticfield on a conductive sphere surface, current density, and anelectromagnetic force. FIGS. 13A to 13C respectively show timevariations in the magnetic field Hx, current density jz, andelectromagnetic force Fy at the gravity center position of a particlenear a conductor surface. In the drawings, a value of an exact solutionis indicated by a solid line, and a calculation value is indicated by abroken line. As shown in the drawings, it has been understood that, ifthe analysis method of this embodiment is used, a calculation result andan exact solution closely match each other.

As described above, the configuration and operation of the analyzer 100according to the embodiments have been described. These embodiments areillustrative, and it will be understood by those skilled in the art thatvarious modification examples may be made to the combination of theconstituent elements or the processes and the modification examplesstill fall within the scope of the invention.

In the embodiments, although a case where the numerical calculation unit120 calculates both the position and speed of the particle has beendescribed, the invention is not limited thereto. For example, as amethod of numerical analysis, like a Verlet method, there is a methodwhich, when calculating the position of the particle, the position ofthe particle is calculated directly from a force applied to the particlewithout explicitly calculating the speed of the particle, and thetechnical spirit of this embodiment may be applied to the method.

It should be understood that the invention is not limited to theabove-described embodiment, but may be modified into various forms onthe basis of the spirit of the invention. Additionally, themodifications are included in the scope of the invention.

What is claimed is:
 1. An analyzer comprising: a magnetic momentapplication unit configured to apply a magnetic moment to a particlesystem defined in a virtual space; a magnetic field calculation unitconfigured to calculate a magnetic physical quantity related to theparticle system including particles, to which the magnetic moment isapplied by the magnetic moment application unit; and a particle statecalculation unit configured to numerically calculate a governingequation, which governs the movement of each particle, using thecalculation result in the magnetic field calculation unit, wherein themagnetic field calculation unit numerically calculates an inductionmagnetic field using induced magnetization induced in each particle dueto a time variation in an external magnetic field and a magnetic fieldobtained by interaction between magnetic moments based on the inducedmagnetization.
 2. The analyzer according to claim 1, wherein themagnetic field calculation unit calculates the induction magnetic fieldusing both of a term representing the induced magnetization and acorrection term of a magnetic field representing interaction betweenparticles having magnetic moments based on the induced magnetization. 3.An analyzer comprising: a magnetic moment application unit configured toapply a magnetic moment to a particle system defined in a virtual space;a magnetic field calculation unit configured to calculate a magneticphysical quantity related to the particle system including particles, towhich the magnetic moment is applied by the magnetic moment applicationunit; and a particle state calculation unit configured to numericallycalculate a governing equation, which governs the movement of eachparticle, using the calculation result in the magnetic field calculationunit, wherein the magnetic field calculation unit numerically calculatesan induction magnetic field in a particle group using an expression foran induction magnetic field derived by gauge transformation using alocal gravity center vector representing the gravity center position ofthe particle group to be analyzed.
 4. The analyzer according to claim 3,wherein the magnetic field calculation unit numerically calculates aninduction magnetic field in the particle group using the expression foran induction magnetic field derived by transforming the vector potentialof the particle system to a gauge described using a vector product ofthe local gravity center vector and the magnetic flux density of theparticle system.
 5. The analyzer according to claim 3, wherein aplurality of particle groups including a first particle group and asecond particle group electrically insulated from each other arearrangeable in the particle system, and the magnetic field calculationunit calculates an induction magnetic field in the first particle groupusing an expression for a first induction magnetic field derived bygauge transformation using a first local gravity center vectorrepresenting the gravity center position of the first particle group andcalculates an induction magnetic field in the second particle groupusing an expression for a second induction magnetic field derived bygauge transformation using a second local gravity center vectorrepresenting the gravity center position of the second particle group.6. The analyzer according to claim 1, wherein the particle system is aparticle system which is renormalized using renormalized moleculardynamics.
 7. The analyzer according to claim 1, wherein the governingequation in the particle state calculation unit has a term depending onthe magnetic moment.
 8. A computer program which causes a computer torealize: a function of applying a magnetic moment to each of particlesof a particle system defined in a virtual space; a function ofcalculating a magnetic physical quantity related to the particle systemincluding the particles, to which the magnetic moment is applied; and afunction of numerically calculating a governing equation, which governsthe movement of each particle, using the calculation result of themagnetic physical quantity, wherein the function of calculating themagnetic physical quantity numerically calculates an induction magneticfield using induced magnetization induced in each particle due to a timevariation in an external magnetic field and a magnetic field obtained byinteraction between magnetic moments based on the induced magnetization.9. A computer program which causes a computer to realize: a function ofapplying a magnetic moment to each of particles of a particle systemdefined in a virtual space; a function of calculating a magneticphysical quantity related to the particle system including theparticles, to which the magnetic moment is applied; and a function ofnumerically calculating a governing equation, which governs the movementof each particle, using the calculation result of the magnetic physicalquantity, wherein the function of calculating the magnetic physicalquantity numerically calculates an induction magnetic field in aparticle group using an expression for an induction magnetic fieldderived by gauge transformation using a local gravity center vectorrepresenting the gravity center position of the particle group to beanalyzed.